Content uploaded by Bojan Šavrič
Author content
All content in this area was uploaded by Bojan Šavrič on Mar 29, 2016
Content may be subject to copyright.
Univerza
v Ljubljani
Fakulteta
za gradbeništvo
in geodezijo
J
amova 2
1000 Ljubljana, Slovenija
telefon (01) 47 68 500
f
aks (01) 42 50 681
f
gg@fgg.uni-lj.si
Kandidat:
Bojan Šavrič
Določitev polinomske enačbe za Naravno
Zemljino kartografsko projekcijo
Diplomska naloga št.: 867
Univerzitetni program Geodezija,
smer Geodezija
Mentor:
doc. dr. Dušan Petrovič
Somentor:
dr. Bernhard Jenny , prof. dr. Lorenz Hurni
Ljubljana, 31. 8. 2011
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. III
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
IZJAVE
Skladno s 27. členom Pravilnika o diplomskem delu UL Fakultete za gradbeništvo in geodezijo,
ki ga je sprejel Senat Fakultete za gradbeništvo in geodezijo Univerze v Ljubljani na svoji 10. seji dne
21. 4. 2010.
Podpisani Bojan Šavrič, izjavljam, da sem avtor diplomske naloge z naslovom: »Določitev polinomske
enačbe za Naravno Zemljino kartografsko projekcijo«. Noben del te diplomske naloge ni bil
uporabljen za pridobitev strokovnega naziva ali druge strokovne kvalifikacije na tej ali na drugi
univerzi ali izobraževalni inštituciji.
Izjavljam, da prenašam vse materialne avtorske pravice v zvezi z diplomsko nalogo na UL Fakulteto
za gradbeništvo in geodezijo. UL FGG ima ob pridobitvi izrecnega pisnega soglasja študenta in
mentorja pravico do javne objave diplomske naloge.
Izjavljam, da dovoljujem objavo elektronske različice na spletnih straneh UL FGG in ETH Zürich,
Institute of Cartography and Geoinformation IKG.
Izjavljam, da je elektronska različica v vsem enaka tiskani različici.
I, Bojan Šavrič, hereby declare that I am the author of the graduation thesis entitled: »Derivation of a
Polynomial Equation for the Natural Earth Projection«.
I confirm that my work may be uploaded on the websites of UL FGG and ETH Zürich,
Institute of Cartography and Geoinformation IKG.
Ljubljana, 31 August 2011
.............................................................
(podpis/signature)
IV Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
THIS PAGE IS INTENTIONALLY LEFT BLANK
TA STRAN JE NAMENOMA PRAZNA
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. V
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
ERRATA
Page Line Error Correction
VI Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. VII
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
BIBLIOGRAPHIC-DOCUMENTALISTIC INFORMATION AND ABSTRACT
UDC: 517.9:528.9(043.2)
Author: Bojan Šavrič
Supervisor: Assist. Prof. Dušan Petrovič, Ph. D.
Co-advisors: Bernhard Jenny, Ph. D., IKG ETH Zürich
Prof. Lorenz Hurni, Ph. D., IKG ETH Zürich
Title: Derivation of a Polynomial Equation for the Natural Earth Projection
Notes: 63 p., 8. tab., 19. fig., 42 eq., 3 ann.
Key words: Natural Earth projection, Flex Projector, Robinson projection, least square
adjustment, LSA, polynomial equation, Newton’s iteration method, forward
projection, inverse projection
Abstract
The Natural Earth projection is a new projection for representing the entire Earth on small-scale maps.
It was designed in Flex Projector, a specialized software application that offers a graphical approach
for the creation of new projections. The original Natural Earth projection defines the length and
spacing of parallels in a tabular form for every five degrees of increased latitude. It is a true pseudo-
cylindrical projection, and is neither conformal nor equal-area. In the original definition, piece-wise
cubic spline interpolation is used to project intermediate values that do not align with the five-degree
grid.
This graduation thesis introduces alternative polynomial equations that are considerably simpler to
compute. The polynomial expression also improves the smoothness of the rounded corners where the
meridians meet the horizontal pole lines, a distinguished mark of the Natural Earth projection which
suggests to readers that the Earth is spherical in shape. An inverse projection is presented. The
formulas are simple to implement in cartographic software and libraries. Distortion values of this new
graticule are not significantly different from the original piece-wise projection.
The development of the polynomial equations was inspired by a similar study of the Robinson
projection. The polynomial coefficients were determined with least square adjustment of indirect
observations with additional constraints to preserve the height and width of the graticule. The inverse
procedure uses the Newton-Raphson method and converges in a few iterations.
VIII Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
BIBLIOGRAFSKO-DOKUMENTACIJSKA STRAN IN IZVLEČEK
UDK: 517.9:528.9(043.2)
Avtor: Bojan Šavrič
Mentor: doc. dr. Dušan Petrovič
Somentorja: dr. Bernhard Jenny, IKG ETH Zürich
prof. dr. Lorenz Hurni, IKG ETH Zürich
Naslov: Določitev polinomske enačbe za Naravno Zemljino kartografsko projekcijo
Obseg in oprema: 63 str., 8. pregl., 19. sl., 42 en., 3 pril.
Ključne besede: Naravna Zemljina projekcija, Flex Projector, Robinsonova projekcija, metoda
najmanjših kvadratov, MNK, polinomska enačba, tangentna metoda, inverzna
enačba
Izvleček
Naravna Zemljina projekcija (ang. The Natural Earth projection) je nova kartografska projekcija za
karte celotnega sveta v majhnih merilih. Zasnovana je bila s pomočjo programa Flex Projector,
iterativnega orodja za grafično oblikovanje in primerjavo kartografskih projekcij. Določena je na
podlagi dolžine in razporeditve paralel za vsakih pet stopinj geografske širine. Vrednosti so podane
tabelarično. Vmesne točke te 5-stopinjske mreže so izvorno določene z interpolacijsko metodo
kubičnih zlepkov. Projekcija je psevdocilindrična in ni niti konformna niti ekvivalentna.
Diplomska naloga predstavlja novo polinomsko enačbo za Naravno Zemljino projekcijo, ki je
enostavnejša za izračun. Obenem enačba izboljšuje tudi gladkost zakrivljenih robov tam, kjer je
stičišče robnega meridiana in pola. Zakrivljeni robovi so posebnost Naravne Zemljine projekcije in
tisti, ki nakazujejo, da je Zemlja okrogla. Polinomske enačbe je mogoče invertirati in so enostavne za
uporabo v kartografskih programih in drugih elektronskih bazah ter aplikacijah. Distorzijski parametri
izboljšane mreže se niso bistveno spremenili in ostajajo skoraj enaki.
Ideja o uporabi polinomskih enačb izhaja iz podobne rešitve za Robinsonovo projekcijo. Vrednosti
posameznih polinomskih koeficientov so bile določene z metodo najmanjših kvadratov. Da smo
ohranili velikost in širino osnovne projekcije, je bila uporabljena posredna metoda izravnave s
funkcijsko odvisnimi spremenljivkami. Inverzna transformacija uporablja tangentno metodo reševanja
enačb, ki kvadratično konvergira h končni rešitvi.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. IX
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
ACKNOWLEDGEMENTS
First of all, I would like to thank my supervisor Assist. Prof. Dr. Dušan Petrovič and co-advisors
Dr. Bernhard Jenny and Prof. Dr. Lorenz Hurni for their expert guidance, encouragement and support
while I was writing my graduation thesis, and their assistance and organization of my exchange study.
I am also thankful to Tom Patterson for his collaboration during my research and graphical evaluation
of the results, and to the employees of Institute of Cartography and Geoinformation, the Swiss Federal
Institute of Technology (ETH) Zürich, who made me feel welcome and offered their help during my
stay there. My words of gratitude go to my parents and brother – thank you for supporting me during
my studies and for accepting all my decisions, even though they were sometimes difficult for you.
I am grateful to Mirjana and others for unconditionally believing in me in challenging times when I
doubted myself. Last but not least, I would like to thank my study colleagues, who always motivated
me. I learned a lot from you, which helped me enter the branch of geodesy.
X Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
ZAHVALA
Zahvaljujem se mentorju doc. dr. Dušanu Petroviču ter somentorjema dr. Bernhardu Jennyju in
prof. dr. Lorenzu Hurniju za njihovo strokovno vodenje, spodbudo in podporo pri moji diplomski
nalogi ter pomoč pri organizaciji študijske izmenjave. Zahvala gre tudi Tomu Pattersonu za
sodelovanje pri nalogi in grafično oceno rezultatov. Hvala zaposlenim na Inštitutu za kartografijo in
geoinformatiko Švicarske tehnične univerze (ETH) Zürich za topel sprejem in pomoč, katere sem bil
deležen v času študija na tej instituciji. Zahvaljujem se tudi staršema in bratu za vso podporo v času
študija ter sprejetje mojih odločitev, pa čeprav so bile te zanje včasih težke. Hvala Mirjani in ostalim,
da ste verjeli vame tudi v trenutkih, ko sem sam podvomil vase. Nenazadnje hvala tudi vsem
študijskim kolegom, ki so mi bili tako ali drugače velika motivacija. Od vseh vas sem se naučil veliko
tega, kar mi je olajšalo vstop v geodetsko stroko.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. XI
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
GENERATION OF THE GRADUATION THESIS
This graduation thesis was written under the supervision and support of my supervisor
Assist. Prof. Dr. Dušan Petrovič, University of Ljubljana and co-advisers Dr. Bernhard Jenny and
Prof. Dr. Lorenz Hurni both ETH Zürich. All work on my thesis, including research and writing, was
done at Swiss Federal Institute of Technology (ETH) Zürich, Institute of Cartography and
Geoinformation IKG, where I obtained all support and guidance I needed. I was studying there as an
exchange student from February till June 2011.
Bojan Šavrič
XII Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
O NASTANKU DIPLOMSKE NALOGE
Diplomsko nalogo sem izdelal pod mentorstvom doc. dr. Dušana Petroviča z Univerze v Ljubljani ter
somentorstvom dr. Bernharda Jennyja in prof. dr. Lorenza Hurnija z ETH Zürich. Celotno delo za
svojo diplomsko nalogo, raziskavo in pisanje sem opravil na Švicarski tehnični univerzi (ETH) v
Zürichu, na Inštitutu za kartografijo in geoinformatiko, kjer sem dobil vso podporo in nasvete, ki sem
jih potreboval. Svoj študij na omenjeni instituciji sem opravljal kot študent na izmenjavi od februarja
do junija 2011.
Bojan Šavrič
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. XIII
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
TABLE OF CONTENTS
BIBLIOGRAPHIC-DOCUMENTALISTIC INFORMATION AND ABSTRACT VII
BIBLIOGRAFSKO-DOKUMENTACIJSKA STRAN IN IZVLEČEK VIII
ACKNOWLEDGEMENTS IX
ZAHVALA X
GENERATION OF THE GRADUATION THESIS XI
O NASTANKU DIPLOMSKE NALOGE XII
TABLE OF CONTENTS XIII
LIST OF TABLES XV
LIST OF FIGURES XVI
1 INTRODUCTION 1
1.1 Problem overview: The Natural Earth projection 1
1.2 Main goal: analytical expression 2
1.3 Second goal: smoothing the corners 2
1.4 Organization of the graduation thesis 2
2 DESIGN AND ORIGIN OF THE NATURAL EARTH PROJECTION 4
2.1 Design of the Natural Earth projection 4
2.2 The characteristics of the Natural Earth projection 6
2.3 The Robinson projection 9
2.4 Analytical expressions for the Robinson projection 10
3 APPLIED NUMERICAL METHODS 12
3.1 Least squares adjustment of indirect observations 12
3.2 Least squares adjustment of indirect observations with additional constraints 13
XIV Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
3.3 The Newton-Raphson iteration method 15
4 A POLYNOMIAL APPROXIMATION OF THE NATURAL EARTH PROJECTION 17
4.1 Derivation of the polynomial to express Cartesian coordinates 18
4.2 Increasing the order of the polynomial 19
4.2.1 The functional model for least squares adjustment 21
4.3 Removing the terms with small contributions 22
4.4 Adding constraints in the least squares adjustment 22
4.5 Improving the smoothness of the rounded corners 24
4.6 Inverting the polynomial equation 26
5 ANALYTICAL EXPRESSION FOR THE NATURAL EARTH PROJECTION –
RESULTS 28
5.1 Forward polynomial expressions 28
5.2 Inverse projection 29
5.3 Comparison of the original and the improved Natural Earth projection 30
6 CONCLUSION 33
7 RAZŠIRJEN POVZETEK DIPLOMSKE NALOGE V SLOVENSKEM JEZIKU 36
8 REFERENCES 50
8.1 Used references 50
8.2 Other references 51
9 APPENDICES 53
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. XV
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
LIST OF TABLES
Table 1: Parameters of the Natural Earth projection (Jenny et al., 2008, page 25). 5
Table 2: Reference variances and maximal residuals of LSA at different polynomial degrees.
The last line shows the values of the LSA after the removal of the terms with small
contribution. Note: Units are millimeters on the map at a scale 1 : 5,000,000, and the
radius of the generating globe equals 6,378,137 meters. 20
Table 3: Coefficients for the polynomial expression of the Natural Earth projection. 29
Table 4: Comparison of the lengths and distances of parallels. 30
Table 5: Distortions for the Natural Earth projection computed with Flex Projector. 32
Tabela P. 1: Vrednost parametrov za Naravno Zemljino projekcijo
(Jenny, et al., 2008, stran 25). 37
Tabela P. 2: Referenčne variance in največji popravki za različne polinomske stopnje po
izravnavi. Zadnja vrstica prikazuje nove vrednosti po odstranitvi členov z
majhnim prispevkom za 12. stopnjo. Opomba: Enote so mm na karti v merilu
1 : 5.000.000, kjer je polmer Zemlje enak 6.378.137 metrov. 43
Tabela P. 3: Koeficienti polinomski enačb za Naravno Zemljino projekcijo. 47
XVI Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
LIST OF FIGURES
Figure 1: High magnification of the rounded corners shows a small edge (in red circle)
at the end of the pole line. 2
Figure 2: The Natural Earth projection applied to the Natural Earth dataset
(Jenny, et al., 2008, page 68). 4
Figure 3: Tissot’s indicatrices for the polynomial Natural Earth projection. 7
Figure 4: Isocols of areal distortion for the Natural Earth projection. 8
Figure 5: Isocols of maximum angular distortion for the Natural Earth projection. 8
Figure 6: The two sets of tabular parameters define the Robinson projection. They are
shown here with horizontal arrows (length of parallels) and vertical arrows
(distance of parallels from the equator). (Jenny et al., 2008, page 13) 9
Figure 7: Pictures A, B, C and D present separated phases of the Newton-Raphson
iteration method. The final solution is (Wikipedia). 16
Figure 8: Fitting curves of different polynomial degrees. Canters and Declair’s polynomial
(4th degree) clearly deviates from the polynomial of degree 12. 20
Figure 9: Screenshots of graphs of both tabular parameters with dependence of the latitude in
Flex Projector. Abscissa has latitude values in degrees units, and ordinate contains
the values of parameters (without units). 26
Figure 10: The graticule with coastlines computed with the polynomial Natural Earth projection. 29
Figure 11: Deviation of the control points on the map at the scale 1: 5,000,000. 31
Figure 12: The original (A) and polynomial (B) Natural Earth projections are overlaid in (C).
Arrows indicate changes in smoothness at the end of the pole line, which is
shortened in (B). 32
Figure 13: Screenshot of the Flex Projector with the polynomial Natural Earth projection. 34
Figure 14: Screenshot of Global Mapper with the polynomial Natural Earth projection
(Global Mapper, 2009). 35
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. XVII
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Slika P. 1: Povečava stičišča projekcij meridiana in pola. Opazen lom je označen
z rdečim krogom. 37
Slika P. 2: Mreža meridianov in paralel Naravne Zemljine projekcije za vsakih 30° geografske
širine in dolžine. Z rdečo barvo sta prikazani tudi koordinatni osi. 38
Slika P. 3: Primerjava izvirne in polinomskih Naravnih Zemljinih projekcij.
Pri uporabljenih polinomih 10., 12., in 14. stopnje razlike niso več opazne. 43
Slika P. 4: Odstopanja kontrolnih točk robnega meridiana na karti merila 1 : 5.000.000. 48
Slika P. 5: Originalna (A) in izboljšana (B) Naravna Zemljina projekcija. Puščice
nakazujejo spremembo gladkosti ob stičišču pola in robnega meridiana. 48
XVIII Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
THIS PAGE IS INTENTIONALLY LEFT BLANK
TA STRAN JE NAMENOMA PRAZNA
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 1
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
1 INTRODUCTION
In this thesis the derivation of a polynomial equation for the Natural Earth projection is presented.
This introduction contains the problem overview and describes the main goal, the determination of the
analytical expression for this projection. The second goal, smoothing the corners at the pole lines, that
improves the studied projection, is presented as well. Finally, the structure of this graduation thesis is
listed.
1.1 Problem overview: The Natural Earth projection
The creation of new map projections by cartographers was difficult mainly because of the general lack
of mathematical expertise and the inability to preview the developed graticule. Until Arthur H.
Robinson presented a graphic approach for developing pseudo-cylindrical projections with tabular
parameters (Robinson, 1974), all projections were developed by analytical equations.
Robinson’s new approach simplified the development of projections, but it does not provide a normal
analytical expression relating longitude / latitude to Cartesian coordinates. Instead of an analytical
equation, his projection was defined by a set of control points. Robinson did not reveal which
interpolation method was used through these points to project intermediate points. For this reason,
there are many different interpolation algorithms for the Robinson projection (Ipbüker, 2005).
An interactive tool for the graphical design and evaluation of map projections that uses Robinson’s
approach is the software applications Flex Projector. According to Jenny et al. (2008), the principal
goal of Flex Projector is to provide tools to expert mapmakers for designing new map projections.
Until now, three projections designed by this application have been published: the A4, the Cropped
Ginzburg VIII and the Natural Earth projection (Jenny et al., 2008, 2010).
Tom Patterson developed the Natural Earth projection in 2007 by using Flex Projector (Jenny et al.,
2008). Since this free, open-source, and cross-platform software application uses Robinson’s
approach, all designed projections – including the Natural Earth projection – do not have normal
analytical expression. Projections are defined by two tabular parameters: the length of parallels and the
distance of parallels from the equator, in sets of 19 control points for each five degrees of positive
latitude. The original Natural Earth projection uses a cubic spline interpolation for each piece of five
degrees. Interpolation is quick to evaluate but it requires a large number of parameters, and its
implementation into geospatial software requires considerable effort. This barrier prevents its
widespread use and because of that, the only software application where the Natural Earth projection
can be computed is Flex Projector, where the projection was designed.
2 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
1.2 Main goal: analytical expression
In order to spread the usage of the Natural Earth projection to other software and projection libraries,
the main goal of this graduation thesis is the derivation of an analytical expression of the projection.
The expression should be a direct transformation of spherical coordinates to Cartesian coordinates. To
simplify the transformation, equations should have a minimum number of parameters, and an inverse
function should also be available. The implementation of the forward and inverse projection must be
simple.
1.3 Second goal: smoothing the corners
The Natural Earth projection is distinguished by its rounded corners, where meridians meet the pole
line. However, corners of the original Natural Earth projection are not completely smooth. At high
magnification small edges can be seen where the pole line ends (Figure 1). In Flex Projector, the five
degree spacing between latitude lines did not provide enough refinement to smooth these corners
completely. The designer of the projection, Tom Patterson, expressed his desire to smooth these
corners.
Figure 1: High magnification of the rounded corners shows a small edge (in red circle) at the end of the pole line.
The second goal of this thesis is to improve the smoothness of the rounded corners where the border
meridians meet the pole lines. The smoothness of these corners should make sure that the edges are
not visible any more.
1.4 Organization of the graduation thesis
The thesis is structured as follows. Chapter 2 introduces the theoretical background of the Natural
Earth projection and different approaches to solve the same challenge for the Robinson projection.
Chapter 3 describes the approximation method using least squares adjustment, and the Newton-
Raphson iteration method for the inverse projection. The development of the polynomial projection is
presented in Chapter 4. It describes the derivation of the polynomial equations and the improvement of
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 3
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
the smoothness of the rounded corners. Chapter 5 presents the results of the thesis consisting in
forward and inverse projections for transforming spherical coordinates to Cartesian / coordinates
and vice versa. Chapter 6 evaluates the improved and the original Natural Earth projections.
Three appendices are added to the thesis. Appendices A and B contain the Matlab code for calculating
the polynomial coefficients for the forward projection, and the function for the inverse projection
written for the purpose of this thesis. Appendix C contains Java code for implementing the Natural
Earth projection in Java applications. This code is included in the Java Map Projection Library
(Jenny, 2011).
This graduation thesis is an expansion of the paper “A polynomial equation for the Natural Earth
projection” by Bojan Šavrič, Bernhard Jenny, Tom Patterson, Dušan Petrovič and Lorenz Hurni,
accepted for the publication in the journal Cartography and Geographic Information Science
(Šavrič et al., accepted for publication).
4
2
DESIGN AND
This
chapter
the design procedure
Natural Earth projection is presented. The secon
Earth projection. Since the Robinson projection was designed with the same graphical approach,
projection does not have an analytical expression. Therefore a short overview of the methods for
find
ing the analytical expressions for the Robinson projection is given
2.1
Design of the Natural Earth projection
The Natural Earth projection was developed by Tom Patterson in 2007 out of
existing projectio
Projector, a freeware application for the interactive design and evaluation of map projections, was
used as
the means for creating the Natural Earth projection. The graphi
Projector allows cartographers to adjust the length, shape, and
new projections in a graphical design process (Jenny and Patterson, 2007).
Figure 2:
The Natural Ea
The Natural Earth projection is an amalgam of the
additional enhancements (
representing small
characteristic
DESIGN AND
ORIGIN OF
chapter
contains
the theoretical background
the design procedure
of the projection
Natural Earth projection is presented. The secon
Earth projection. Since the Robinson projection was designed with the same graphical approach,
projection does not have an analytical expression. Therefore a short overview of the methods for
ing the analytical expressions for the Robinson projection is given
Design of the Natural Earth projection
The Natural Earth projection was developed by Tom Patterson in 2007 out of
existing projectio
ns for displaying physical data on small
Projector, a freeware application for the interactive design and evaluation of map projections, was
the means for creating the Natural Earth projection. The graphi
Projector allows cartographers to adjust the length, shape, and
new projections in a graphical design process (Jenny and Patterson, 2007).
The Natural Ea
rth projection
The Natural Earth projection is an amalgam of the
additional enhancements (
representing small
-
scale physical data on world
characteristic
(Jenny et al., 2008)
Šavrič
ORIGIN OF
THE NATURAL EARTH PR
the theoretical background
of the projection
Natural Earth projection is presented. The secon
Earth projection. Since the Robinson projection was designed with the same graphical approach,
projection does not have an analytical expression. Therefore a short overview of the methods for
ing the analytical expressions for the Robinson projection is given
Design of the Natural Earth projection
The Natural Earth projection was developed by Tom Patterson in 2007 out of
ns for displaying physical data on small
Projector, a freeware application for the interactive design and evaluation of map projections, was
the means for creating the Natural Earth projection. The graphi
Projector allows cartographers to adjust the length, shape, and
new projections in a graphical design process (Jenny and Patterson, 2007).
rth projection
applied to the Natural Earth dataset (
The Natural Earth projection is an amalgam of the
additional enhancements (
Figure 2
). These two pro
scale physical data on world
(Jenny et al., 2008)
.
The Kavraiskiy VII projection exaggerates the size of high latitude
Šavrič, B. 2011. Derivation of a
Grad. Th. -
University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
THE NATURAL EARTH PR
the theoretical background
of the projection
in Flex Projector
Natural Earth projection is presented. The secon
Earth projection. Since the Robinson projection was designed with the same graphical approach,
projection does not have an analytical expression. Therefore a short overview of the methods for
ing the analytical expressions for the Robinson projection is given
Design of the Natural Earth projection
The Natural Earth projection was developed by Tom Patterson in 2007 out of
ns for displaying physical data on small
Projector, a freeware application for the interactive design and evaluation of map projections, was
the means for creating the Natural Earth projection. The graphi
Projector allows cartographers to adjust the length, shape, and
new projections in a graphical design process (Jenny and Patterson, 2007).
applied to the Natural Earth dataset (
The Natural Earth projection is an amalgam of the
). These two pro
scale physical data on world
The Kavraiskiy VII projection exaggerates the size of high latitude
, B. 2011. Derivation of a
P
olynomial
University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
THE NATURAL EARTH PR
the theoretical background
of the
studied projection and related work
in Flex Projector
is
described
Natural Earth projection is presented. The secon
d section describes the characteristics of the Natural
Earth projection. Since the Robinson projection was designed with the same graphical approach,
projection does not have an analytical expression. Therefore a short overview of the methods for
ing the analytical expressions for the Robinson projection is given
The Natural Earth projection was developed by Tom Patterson in 2007 out of
ns for displaying physical data on small
-
scale world maps (
Projector, a freeware application for the interactive design and evaluation of map projections, was
the means for creating the Natural Earth projection. The graphi
Projector allows cartographers to adjust the length, shape, and
new projections in a graphical design process (Jenny and Patterson, 2007).
applied to the Natural Earth dataset (
The Natural Earth projection is an amalgam of the
Kavra
i
). These two pro
jections most closely fulfilled the requirement for
scale physical data on world
maps,
however
The Kavraiskiy VII projection exaggerates the size of high latitude
olynomial
E
quation for the Natural Earth
University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
THE NATURAL EARTH PR
OJE
CTION
studied projection and related work
described
,
and the definition of the original
d section describes the characteristics of the Natural
Earth projection. Since the Robinson projection was designed with the same graphical approach,
projection does not have an analytical expression. Therefore a short overview of the methods for
ing the analytical expressions for the Robinson projection is given
at
The Natural Earth projection was developed by Tom Patterson in 2007 out of
scale world maps (
Projector, a freeware application for the interactive design and evaluation of map projections, was
the means for creating the Natural Earth projection. The graphi
Projector allows cartographers to adjust the length, shape, and
spacing
new projections in a graphical design process (Jenny and Patterson, 2007).
applied to the Natural Earth dataset (
Jenny,
i
skiy
VII and Robinson projections, with
jections most closely fulfilled the requirement for
however
,
each had at least one undesirable
The Kavraiskiy VII projection exaggerates the size of high latitude
quation for the Natural Earth
University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
CTION
studied projection and related work
and the definition of the original
d section describes the characteristics of the Natural
Earth projection. Since the Robinson projection was designed with the same graphical approach,
projection does not have an analytical expression. Therefore a short overview of the methods for
at
the end of this chapter.
The Natural Earth projection was developed by Tom Patterson in 2007 out of
his
dissatisfaction with
scale world maps (
Jenny et al.,
Projector, a freeware application for the interactive design and evaluation of map projections, was
the means for creating the Natural Earth projection. The graphi
cal user interface in Flex
of parallels and meridians of
new projections in a graphical design process (Jenny and Patterson, 2007).
Jenny,
et al., 2008
VII and Robinson projections, with
jections most closely fulfilled the requirement for
each had at least one undesirable
The Kavraiskiy VII projection exaggerates the size of high latitude
quation for the Natural Earth
P
rojection
University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
studied projection and related work
.
Firstly,
and the definition of the original
d section describes the characteristics of the Natural
Earth projection. Since the Robinson projection was designed with the same graphical approach,
projection does not have an analytical expression. Therefore a short overview of the methods for
the end of this chapter.
dissatisfaction with
Jenny et al.,
2008). Flex
Projector, a freeware application for the interactive design and evaluation of map projections, was
cal user interface in Flex
of parallels and meridians of
et al., 2008
, page 68).
VII and Robinson projections, with
jections most closely fulfilled the requirement for
each had at least one undesirable
The Kavraiskiy VII projection exaggerates the size of high latitude
rojection
.
University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Firstly,
and the definition of the original
d section describes the characteristics of the Natural
Earth projection. Since the Robinson projection was designed with the same graphical approach,
this
projection does not have an analytical expression. Therefore a short overview of the methods for
dissatisfaction with
2008). Flex
Projector, a freeware application for the interactive design and evaluation of map projections, was
cal user interface in Flex
of parallels and meridians of
VII and Robinson projections, with
jections most closely fulfilled the requirement for
each had at least one undesirable
The Kavraiskiy VII projection exaggerates the size of high latitude
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 5
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
areas, resulting in oversized representation of polar regions. The Robinson projection, on the other
hand, has a height-to-width ratio close to 0.5, resulting in a slightly too wide graticule with outward
bulging sides and too much shape distortion near map edges.
Creating the Natural Earth projection required
three major adjustments: Firstly, starting from the
Robinson projection, its vertical extension slightly
increased the height-to-width ratio from 0.5072 to
0.52 to give it more height. Secondly, using the
Kavraiskiy VII as a template, the parallels were
slightly increased in length. And thirdly, the length
of the pole lines was decreased by a small amount
to give the corners at pole lines a rounded
appearance. Designing the Natural Earth
projection in this way required trial-and-error
experimentation and visual assessment of the
appearance of continents in an iterative process
(Jenny et al., 2008).
The shape of the graticule of any projection
designed with Flex Projector is defined by tabular
sets of parameters. For the Natural Earth
projection, two parameter sets are used for specifying (1) the relative length of the parallels, and
(2) the relative distance of parallels from the equator (Table 1). Equation 1 defines the original Natural
Earth projection, transforming spherical coordinates to Cartesian / coordinates
(Jenny et al., 2008, 2010):
, , (Equation 1)
, ,
where:
and are projected coordinates,
is the radius of the generating globe,
is an internal scale factor,
is the relative length of the parallel at latitude , with
, for the equator,
and with the slope of is 63.883° at the poles,
Table 1: Parameters of the Natural Earth projection
(Jenny et al., 2008, page 25).
Latitude Length of
Parallels
Distance of
Parallels from
Equator
0 1 0
5 0.9988 0.062
10 0.9953 0.124
15 0.9894 0.186
20 0.9811 0.248
25 0.9703 0.31
30 0.957 0.372
35 0.9409 0.434
40 0.9222 0.4958
45 0.9006 0.5571
50 0.8763 0.6176
55 0.8492 0.6769
60 0.8196 0.7346
65 0.7874 0.7903
70 0.7525 0.8435
75 0.716 0.8936
80 0.6754 0.9394
85 0.627 0.9761
90 0.563 1
Height / width 0.52
Scale 0.8707
6 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
is the vertical distance of the parallel at latitude from the equator, with
and
with for the pole lines, and for the equator,
is the longitude with and
is the height-to-width ratio of the projection.
Arthur H. Robinson proposed the structure of Equation 1 and the associated graphical approach to the
design of small-scale map projections when he developed his eponymous projection (Robinson, 1974).
In making the Natural Earth projection, Jenny et al. (2010) provide numerical values for the tabular
parameters that define and in Equation 1 for each five-degree value (Table 1). For intermediate
spherical coordinates that do not align with the five-degree grid, values for and need to be
interpolated. The Flex Projector application uses a piece-wise cubic spline interpolation, with each
piece of the spline curve covering five degrees. While this type of interpolation can be evaluated
quickly, it is relatively intricate to program and requires a large number of parameters – factors that
are likely to impede the widespread implementation of the Natural Earth projection in geospatial
software. Seeking greater efficiency, the connection between parameters in Table 1 and spherical
coordinates must be found in a compact analytical expression that approximates Equation 1.
2.2 The characteristics of the Natural Earth projection
The result of the procedure described in the previous section, the Natural Earth projection (Figure 2),
is a true pseudo-cylindrical projection, i.e., a projection with regularly distributed meridians and
straight parallels (Snyder, 1993, p. 189). The projection of the equator line defines the axis, and the
central meridian defines the axis (Figure P. 2, on page 38). The graticule is symmetric about the
central meridian and the equator. The equator is 0.8707 times longer than the circumference of a
sphere of equal area. The central meridian is a straight line 0.52 times longer than the equator. Other
meridians are equally spaced elliptical arcs and do not intersect the parallels at straight angles. They
are concave toward the central meridian. The corners where pole lines and bounding meridians meet
have a rounded appearance, which is a distinguished characteristic among pseudo-cylindrical
projections. Other meridians have a less rounded appearance, the closer they are to the central
meridian the more they resemble a straight line.
As a compromise projection, the Natural Earth projection is neither a conformal nor equal area, but its
distortion characteristics are comparable to, other well known projections. Its distortion values fall
somewhere between those of the Kavraiskiy VII and Robinson projection that were used in the design
procedure. Similar to all compromise projections, the Natural Earth projection also exaggerates the
size of high latitude areas (Jenny et al., 2008). Figures 3, 4 and 5 illustrate these characteristics.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 7
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
In Figure 3 the Tissot’s indicatrices are presented for each 30 degrees. With larger distance from the
equator, the area of indicatrices increases. The largest indicatrices on parallels of 60 degrees indicate
that the projection exaggerates the size of high latitude areas. Since the size of indicatrices is
enormous at pole lines, they are not presented in this figure. The axes of the indicatrices do not
coincide with the directions of parallels and meridians, except at the equator and the central meridian.
Since the projection is symmetric the Tissot indicatrices are also symmetric.
Figure 3: Tissot’s indicatrices for the polynomial Natural Earth projection.
Tissot’s indicatrices are not the only way to present the projection’s characteristics. Instead of showing
them on the map, different local measures of one particular distortion parameter, such as maximum
areal distortion or maximum angular distortion, can be mapped by lines of equal distortion, called
isocols (Figure 4 and 5).
Figure 4 presents isocols of areal distortion for the Natural Earth projection. The increased density of
isocols at high latitudes also indicates an enlargement of those areas. Along the equator the areal
distortion equals 0.87699. This value of areal distortion is the smallest for the Natural Earth projection.
In Figure 4 the last isoline with the value of 5.0 can hardly be seen, as it is very close to the pole line.
Areal distortion increases with latitude and does not change with longitude. All isocols of areal
distortion are therefore parallel to the equator. Areal distortion is computed with with
and the scale factors along the principal directions at position on the sphere.
8 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Figure 4: Isocols of areal distortion for the Natural Earth projection.
In Figure 5, the isocols of maximum angular distortion are presented. Their general pattern is common
to all pseudo-cylindrical projections. Angular distortion is moderately near the equator and increases
towards the edges of the graticule. It grows with an increased latitude. The last shown isoline is at
120 degrees, at the pole line, the distortion is 180 degrees. As for Tissot’s indicatrices, the isocols of
areal and maximum angular distortion are also symmetric.
Figure 5: Isocols of maximum angular distortion for the Natural Earth projection.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 9
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
2.3 The Robinson projection
There are considerable similarities between the Natural Earth projection and the Robinson projection.
The Robinson projection is a pseudo-cylindrical projection, neither conformal nor equal area. The
graticule is symmetric about the central meridian and the equator. The equator is 0.8487 times longer
than the circumference of a sphere of equal area. The central meridian is a straight line 0.5072 longer
than the equator. Other meridians are equally spaced elliptical arcs, and concave toward the central
meridian (Robinson, 1974, Ipbüker, 2004).
The Robinson projection was designed by Arthur H. Robinson in 1974 for the Rand McNally
Company. Robinson and Patterson used an identical approach for the design of their true pseudo-
cylindrical projections. Both defined their projection by adjusting the appearance of the projected five-
degree graticule in an iterative process – Robinson sketching the graticule with pen and paper, and
Patterson fine-tuning it in Flex Projector. The identical procedure results in the same problems: both
projections do not have normal analytical expressions.
Figure 6: The two sets of tabular parameters define the Robinson projection. They are shown here with
horizontal arrows (length of parallels) and vertical arrows (distance of parallels from the equator)
(Jenny et al., 2008, page 13).
10 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
2.4 Analytical expressions for the Robinson projection
In the past, various authors tackled the problem of finding an analytical expression for the Robinson
projection. Since the two projections are closely related, this section reviews existing mathematical
models of the Robinson projection.
There are two general approaches to the mathematical modeling of graphically defined projections:
(1) interpolation and (2) approximation. Both approaches have been applied to Robinson’s projection.
Interpolating methods use a function that passes exactly through the reference points. Ipbüker
(2004; 2005) presents a method based on multiquadric interpolation for the forward and the inverse
projection. Others use interpolating methods for finding continuous expressions of lφ and dφ in
Equation 1. For example, Snyder (1990) applies the central-difference formula by Stirling; Ratner
(1991), Bretterbauer (1994), and Evenden (2008) use cubic spline interpolation (which is also used in
Flex Projector); and Richardson (1989) reports that Robinson applied the Aitken interpolation scheme.
A disadvantage of the mentioned interpolating methods is the large number of parameters required
(more than 40 for the Robinson projection), and their relatively difficult implementation. For this
reason they are not explored further here.
Approximating curves with parametric expressions that do not exactly replicate the original projection
are an acceptable alternative, if deviations to the approximated values are small. Canters and Decleir
(1989) present two polynomial equations (Equation 2) for approximating the Robinson projection. For
the coordinates they use even powers up to the order four, and for the coordinates odd powers up
to the order five. Each expression contains three coefficients, and the constants , and of Equation
1 are integrated with and . Their solution contains only six parameters, and is fast and simple to
compute.
, (Equation 2)
,
where:
and are the projected coordinates,
and are the latitude and longitude in radians,
is the radius of the generating globe,
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 11
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
with the coefficients being:
, , ,
, and .
A similar approach is proposed by Beineke (1991; 1995). For he suggests a polynomial with even
degrees up to the sixth order, and for he proposes an exponential approximation with a real number
exponent (Equation 3) (Beineke, 1991). This approach uses a total of eight parameters to approximate
the Robinson projection. However, evaluating an exponential function with a real number exponent is
slow. A test with the Java programming language, for example, shows that Beineke's exponential
approximation is more than ten times slower to evaluate than a polynomial, such as the one by
Canters and Decleir.
(Equation 3)
where:
and are the projected coordinates,
and are the latitude and longitude in radians,
with the coefficients being:
, , ,
, , , and
.
The approximating curves by Canters and Decleir, as well as Beineke, use a smaller number of
parameters, and they are considerably simpler to program than the interpolating methods. Polynomial
equations are the best in terms of computation speed and code simplicity, but higher-order terms might
be necessary to minimize deviations from the original curve. However, polynomial approximations
sometimes suffer from undulations if the maximum degree is too high, which must be avoided for a
graticule to appear smooth. A drawback of polynomial equations is the difficulty of finding inverse
equations that transform from projected coordinates to spherical coordinates. Indeed, an
analytical inverse does not generally exist for higher-order polynomial equations. In order to transform
Cartesian coordinates in to spherical coordinates, numerical approximation methods are necessary,
such as the bisection or the Newton-Raphson root finding algorithm.
12 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
3 APPLIED NUMERICAL METHODS
This chapter presents two approximation methods using least squares adjustment (LSA). The first
method is LSA of indirect observations and the second approach is a modified LSA of indirect
observations with functionally dependent parameters presented by Mikhail and Ackerman (1976).
LSA of indirect observations is used to determine the degree of the polynomial. The value of residuals
and the reference variance are applied to estimate the results between different approximations.
Similar to all approximation methods, this procedure does not provide a curve passing exactly through
all points. Using the second method, parameter constraints are added in the functional model of LSA,
considering geometric characteristics that are to be preserved. With this method, the values of
parameters were calculated. The applied functional models in LSA procedure are presented in
Chapter 4. The last section in this chapter presents the Newton-Raphson method used for inverting the
polynomial projection.
3.1 Least squares adjustment of indirect observations
An adjustment of indirect observations is one of the two classical cases of LSA. The adjustment is
performed with parameters and observations but with the restriction that only one observation is used
in each condition equation. Therefore, the number of condition equations is the same as the number of
observations (). In each condition equation the observation is expressed with parameters (). The
general linear functional equations in this matrix form are:
or , (Equation 4)
where:
is the vector of observations,
is the vector of residuals,
and are the vectors of parameters,
and are the coefficient matrices of , and
and are vectors of constants.
Both matrix forms in Equation 4 present an identical mathematical model, and for the functional
model used for the Natural Earth projection they provide the same results. The second form is more
widespread and applied in this case. It is therefore presented here.
Usually the number of condition equations is larger than the numbers of parameters (). For this
reason there is some redundancy or degree of freedom (), and no unique solution exists for the system
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 13
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
in Equation 4. A unique least squares solution is obtained by adding the basic criterion:
, where presents the weight matrix of the observations. With this
criterion, the solution is given in Equation 5.
, (Equation 5)
,
where:
is the reduced normal coefficient matrix, and
is the reduced normal equations constant vector.
If LSA uses linear functions, no initial guesses are required. However, some conditions can only be
expressed in non-linear manner. The parameters in the non-linear case are:
, where is a
vector of approximations and contains only the corrections of these approximations. The coefficient
matrix contains partial derivatives of the condition equations with respect to all parameters. The
partial derivatives are computed from initial guesses in vector . In the linear case, these partial
derivatives are coefficients. To express non-linear conditions, certain means of linearization are used;
e.g. Taylor’s series (using the zero and first order terms and neglecting others). With the non-linear
expression, Equation 5 is an iterative procedure where corrections of parameters are iteratively
improved until they fulfill the required accuracy.
(Equation 6)
Equation 6 presents an unbiased estimate of the reference variance, computed from the quadratic form
of the residuals, where is the redundancy. For the LSA of indirect observations, the redundancy is
defined as a difference between the number of condition equations () and the number of unknown
parameters (): .
3.2 Least squares adjustment of indirect observations with additional constraints
In the functional model of Equation 4, constraint equations can be added and defined only by
parameters (Equation 7). This new group of extra equations implies that some of the parameters are
functionally dependent. The parameter constraints are used when some of them must fulfill certain
additional relationships from either a geometric or physical characteristic of the functional model.
Since the constraint equations are expressed with the parameters from the adjustment, the number of
14 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
dependent parameters () or their equations must be lower than the number of parameters (: .
With the addition of constraint equations the redundancy is changed (Mikhail and Ackerman, 1976).
For the LSA of indirect observations, the sum of conditions and constraints is equal to the redundancy
plus the total numbers of all dependant and independent parameters: .
(Equation 7)
where:
is the coefficient matrix of and
is vector of constants.
Equation 7 expresses the relationship between the functionally dependent parameters that must be
included in LSA (Equation 5). This step is presented in Equation 8, where the two systems in
Equations 4 and 7 are solved together. The first row in Equation 8 is a normal LSA of indirect
observations. The results are parameters without the constraints. In the second row of Equation 8, the
corrections for the parameters are calculated. In the third row, the vector is computed containing
the coefficients of the polynomial approximation. And finally, the vector of residuals is computed.
The polynomial in fulfills all additional parameter constraints expressed in Equation 7, and
minimizes the correction of all observations.
(Equation 8)
As the conditions are linear, no iterations are needed to solve the functional model, and no initial
guesses are required for the unknown parameters. All constraints can be expressed with functionally
dependent parameters. Even if the conditional equations are linear, non-linear constraints can be used.
In this case, matrix contains partial derivatives of the constraint equations with respect to all
parameters in vector , calculated from parameter values in vector . However, non-linear constraints
can only partially be fulfilled. If the conditions are originally non-linear, constraints are expressed with
the corrections of initial guesses for parameters. The form of those constraints depends on the values
of the initial guesses.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 15
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
3.3 The Newton-Raphson iteration method
Many cartographic projections do not have an equation for the inverse projection from projected
coordinates to spherical ones . For the inverse projection different approximation and
iteration methods are used, such as the bisection method, Euler method or the Newton-Raphson root
finding algorithm.
The Newton-Raphson algorithm is a numerical method for finding successively better approximations
to the roots or zeroes of a real-valued function (Wikipedia), and is commonly used for the numerical
calculation of non-linear equations. Equation 9 shows the general form of the Newton-Raphson
algorithm.
, (Equation 9)
where:
and are a given function and its derivative,
and are the previous and the next solution of the given function, and
n and the steps of the iterative process.
The algorithm starts with the first initial guess and computes an improved approximation of the
solution: . In the next step, this new approximation is used instead of , and is again closer to
the solution. The procedure continues until the difference between approximated values is sufficiently
small, . The iteration stops at , which is the final solution. When the first initial guess is close
enough to the final solution and the derivative of the given function has the same sign (just positive or
just negative, but not zero) as in the point of the solution, the algorithm converges very quickly and its
convergence is quadratic.
Geometrically, this method does not search the intersection of the graph with the x-axis, but the
intersection point of the tangent line to the graph with the x-axis in the start point .
From this geometrical meaning the alternative name for this method is tangent iteration method
(Bronštejn et al., 1974).
Figure 7 demonstrates this method. As an initial guess the value is used, from which the tangent
and intersection of the tangent line with the x-axis is computed. In step B, an approximation of the
root, is used, and an improved approximation is computed. The final result in step D is .
16 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Figure 7: Pictures A, B, C and D present separated phases of the Newton-Raphson iteration method. The final
solution is (Wikipedia).
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 17
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
4 A POLYNOMIAL APPROXIMATION OF THE NATURAL EARTH PROJECTION
In a trial-and-error process, a polynomial approximation with a minimum number of terms was
determined for the original Natural Earth projection. Polynomials of varying degrees and different
number of terms were selected and their coefficients were computed using the method of least squares
with constraints. Two criteria were used to evaluate variants developed with this iterative trial-and-
error procedure. First, the number of polynomial terms and the number of multiplications required to
evaluate the equation need to be minimized. This criterion is important for simplifying the
programming of the equations. It is also relevant for accelerating computations, for example, for web
mapping application that projects maps on the fly using JavaScript or other interpreted programming
languages that are comparatively slow. The second criterion aims at minimizing the absolute
differences between the original projection and the approximated projection. Differences should be
minimal throughout the entire projection.
When designing the original Natural Earth projection, special focus was given to the smoothness of
the rounded corners where the meridians meet the horizontal pole lines. It was found that the graphical
tools and the cubic spline interpolation in Flex Projector do not provide sufficient control for defining
rounded corners with adequate smoothness. The development of a polynomial approximation provided
the possibility to further improve this distinguishing characteristic of the Natural Earth projection,
which was also a wish of the designer of the original projection. The new polynomial form of the
projection therefore deliberately deviates from the original projection by adding curvature to the
corners. It has to be noted that the changes applied to the smoothness of the corners are entirely
esthetic and satisfy the authors’ sensibilities. They result in a subjective improvement that cannot be
evaluated with objective criteria. Furthermore, changes are not applied to improve the projection’s
distortion characteristics.
The development of the polynomial approximation was assessed by Tom Patterson, the projection
designer. Since his wish was to improve the rounded corners, his graphical evaluation has more
significance than the absolute differences between the original projection and the approximated
projection. Approximations with different polynomial degrees were evaluated during this process to
minimize the absolute difference between the approximated and the original projection. For each
approximation the reference variance was computed from the residuals. When the variance did not
change much with an increased polynomial degree, the approximation was considered sufficient. The
polynomial equations were then implemented in Flex Projector for graphical review of the results and
also for the comparison with the original projection. This graphical evaluation was made by the
designer. By using this graphic approach for the evaluation of the new projection, the results reflect
the subjective evaluation of the designer. The polynomial approximation of the Natural Earth
18 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
projection (hereafter the term polynomial Natural Earth projection is used for this approximation) was
developed in six separated phases, presented in the following subsections. Appendix A provides the
Matlab code, which is used for the evaluation of the polynomial coefficients.
4.1 Derivation of the polynomial to express Cartesian coordinates
Each Cartesian coordinate, and , can be expressed by two spherical coordinates, and . The
general form of a polynomial with two variables (Equation 10) has
terms, where is
the degree of the polynomial. In the case of 3 degrees, 20 parameters are needed to express both
coordinates on the map plane. Some values of those parameters are equal or close to zero because of
the characteristics of the projection. Since these terms are not needed, the general form can be
simplified.
(Equation 10)
To simplify the terms in Equation 10, the following considerations given by Canters (2002) for
deriving new graticules with polynomials were taken into account:
(1) The Natural Earth projection is symmetric about the and -axis;
(2) It has straight but not equally spaced parallels;
(3) The parallels are equally divided by meridians.
The equator of the graticule represents the -axis and the central meridian the -axis (Figure P. 2,
on page 38). Longitude changes with the coordinate. The coordinate has the same sing as
latitude . When the projection is symmetric about the -axis, even powers of preserve the sign of
the coordinate as provided by latitude , and odd powers of preserve the sign for the coordinate.
Symmetry about the -axis is obtained in a similar way. Even powers of preserve the sign of the
coordinate as provided by longitude and odd powers of preserve the sing for the coordinate.
By taking the symmetry relative to both axes into consideration, most of the terms in Equation 10 are
eliminated. For the coordinate, all coefficients of even powers of and odd powers of are
removed, and for the coordinate all terms with odd powers of and all even powers of are
removed. Equation 11 presents this removal for the polynomial of 3rd degree (Canters, 2002).
(Equation 11)
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 19
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
The second characteristic, straight but not equally spaced parallels, allows for making the coordinate
a function of latitude only, since the parallels do not change with longitude. The unequal distribution
of parallels of the Natural Earth projection has to be expressed as a non-linear function of latitude
(Equation 12). A linear expression would result in equally spaced parallels (Canters, 2002).
The last mentioned characteristic, the maintenance of the equal spacing of meridians, forms the
function for coordinates. This characteristic is dependent on longitude, and for true pseudo-
cylindrical projection a linear expression should be used (Canters, 2002). Therefore the expression of
coordinate contains only the terms with linear longitude (Equation 12).
(Equation 12)
As can be seen in Equation 12, for the polynomial of third degree only 4 coefficients remain, which is
much less than the 20 terms in Equation 11. If these parameters were not removed from the
expression, LSA would determine them as equal or close to zero. Therefore in the next phases for the
coordinate only even powers of latitude , multiplied with the longitude are used, and
coordinate contains only odd powers of latitude .
4.2 Increasing the order of the polynomial
Once the form of the polynomials is determined, finding an appropriate degree is the next step. If the
degree is too low, the curve deviates from the control points with big residuals. If the degree is too
high, the curve waves through these points, although the residuals are relatively smaller. Different
variants were compared on the basis of the reference variance, calculated from the quadratic form of
residuals, and by a graphical comparison of the approximated and the original curves (Figure 8). A
low polynomial degree makes the approximate curve more dissimilar to the original one. For example,
in Figure 8 the curve of degree four deviates from the original because the degree is too low. With a
higher degree (degree twelve in Figure 8) the curve is closer to the original and the reference variance
becomes smaller. To achieve a better curve fitting for the Natural Earth projection, the degree of
polynomials in Equation 12 is increased to 12 (Equation 13).
(Equation 13)
20 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Figure 8: Fitting curves of different polynomial degrees. Canters and Declair’s polynomial (4th degree) clearly
deviates from the polynomial of degree 12.
In Table 2 the effect of increasing the order of the polynomial is presented. For each tabular parameter
different polynomial degrees were used in LSA, with the reference variance (Equation 6) and the
maximal residual used for the first evaluation. The length of the parallels and their distribution were
approximated separately, and they are presented in Table 2. The degree of polynomials in the
functional model was increased by one term in each step. Since the reference variance and the
maximal residuals of the 10th and 12th degree do not change much, the curve could start undulating
between control points. The maximal residuals were about 1 millimeter at a scale of 1: 5,000,000. This
scale presents a map with dimensions of 3.489 × 1.814 meters. The conclusion was that this solution is
accurate enough for the graphical evaluation. The polynomial functions were implemented in the Flex
Projector application. The used functional model is presented in the next sub-subsection.
Table 2: Reference variances and maximal residuals of LSA at different polynomial degrees. The last line shows
the values of the LSA after the removal of the terms with small contribution. Note: Units are millimeters on the
map at a scale 1 : 5,000,000, and the radius of the generating globe equals 6,378,137 meters.
Polynomial
degree
The length of parallels (
) The distance of parallels (
)
[mm] max (v) [mm]
[mm] max (v) [mm]
6
th
7.322 14.701 2.564 5.483
8
th
2.328 4.556 1.323 2.447
10
th
0.494 1.186 0.499 0.760
12
th
0.406 1.173 0.291 0.813
14
th
0.359 0.911 0.186 0.476
*12
th
0.406 1.081 0.287 0.823
0.5500
0.6000
0.6500
0.7000
0.7500
0.8000
0.8500
60 65 70 75 80 85 90
The length of parallels lφ
Latitude φ[°]
Comparison of original and approximated curves of
the Natural Earth projection
The Natural Earth projection Polynomial of degree 4 Polynomial of degree 12
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 21
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
4.2.1 The functional model for least squares adjustment
To determine the desired polynomial orders, LSA of indirect observations was used for computing the
necessary residuals for evaluation. For each LSA the functional model must be composed first. The
original Natural Earth projection is defined by 37 control points distributed over the complete lateral
meridian (with and a control point every 5 degrees). The functional model of the
LSA is given in Equation 14, which is derived from Equations 1 and 13:
, (for )
. (Equation 14)
The coefficients of the two polynomial expressions in Equation 14 are unknown parameters, and the
polynomial powers of latitude are known coefficients for the LSA. Both systems of equations were
adjusted separately (Equation 14), with n denoting the number of rows in the coefficients matrix A
(), and the number of parameters which is increased. The separate matrix or vector terms to
express the length of parallels () and the vertical distance of the parallel from the equator () are
introduced in Equations 15 and 16.
(Equation 15)
(Equation 16)
The index 1 presents the approximation of , and index 2 the approximation of . Vectors and
contain the parameters, i.e. the unknown polynomial coefficients. In this particular case, no
parameters are included in the matrices and , which results in a linear system that can be solved
using the method of least squares in Equation 5. The vectors and represent the minimized
residuals after adjustments. and are used for evaluating the reference variance of the model, and
the differences between the original and the approximated graticule. The vectors and include
or , multiplied by the constant factors , and as in Equation 14. Since this is a linear model, no
initial guess is required for the parameters. All observations in vectors and have the same
22 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
relevance. Therefore the weight matrices ( and ) are equal to the identity matrices in both cases.
The weight matrix can be removed from Equations 5, 6 and 8, and calculations are simplified.
4.3 Removing the terms with small contributions
Reducing the number of terms in the polynomials accelerates computations. The values of the
polynomial coefficient indicate a possibility to remove some of the polynomial terms.
The results of a LSA of polynomials with 12th degree indicate a possibility to remove the sixth and
eighth power terms in the equation that expresses coordinates. As for the equation of coordinate,
the fifth power term has the smallest value. All three terms were removed in this step, and LSA with
the new functional model (without these terms) was computed. Removing the terms did not
considerably change the graphical appearance. Moreover, Table 2 shows that reference variances and
maximal residuals are even better. Therefore the removal is acceptable.
The equations by Robinson (Equation 1) can be expressed with these reduced polynomials. By
multiplying the polynomials (Equation 13) with the radius of the generating globe, the polynomials
express the multiplied tabular parameters and , including the constant factors , and in
Equation 17. This form of equations can be accepted as the final form, but coefficients included in
polynomials must be determined with additional constraints.
(Equation 17)
4.4 Adding constraints in the least squares adjustment
The graticule of the Natural Earth projection provided by polynomial equations is not exactly the same
size as the original graticule. The deviations are caused by the LSA approximation method. Residuals
from LSA present this discrepancy for each coordinate of the control points. In Figure 8 this
discrepancy is evidently represented for the length of parallels. At 90 degrees of latitude this change
can be seen by comparing the original Natural Earth projection with an approximated curve of the
fourth polynomial degree. To preserve the size and proportions of the graticule the length of the
equator and the distance between the pole line and the equator have to remain the same.
Hence, when estimating the polynomial coefficients with the method of least squares, two additional
constraints were added to bring the polynomial graticule to the exact same size as the original
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 23
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
graticule: (1) the length of the parallel at 0 degrees must be 1; (2) the relative distance between the
equator and the parallel at 90 degrees must be 1. Both constraints are expressed with parameters,
which is possible because the expressions in Equation 17 are linear.
For computing the polynomial coefficients of the relative length , 37 control points are used
covering the whole range of possible latitude values between
and
with a distance of five
degrees between each pair of control points. The symmetrical arrangement of the control points
around the equator guarantees a continuously differentiable function.
The first additional constraint (1) for the length can be derived from Equations 14 and 17. This
functional dependence is described in the following equation:
(Equation 18)
The first coefficient was forced to equal the value of the internal scale factor (Equation 1), which
ensures that the length of the equator remains the same. Since this constraint is the only one for the
length , it must be written separately with the matrix and vector for Equation 7. With the
matrix and vectors from Equation 15 it forms the functional model for the length of parallels.
(Equation 19)
The constraint (2) is applied to the distance . The fixed distance of the parallel at 90 degrees is
expressed in a similar way as the constraint (1) in Equation 18, and it must also be present in a
separated matrix form (Equation 21).
(Equation 20)
(Equation 21)
Matrices and vectors in the equations 15, 16, 19 and 21 form two functional models, whose results
provide the polynomial graticule of the Natural Earth projection. For the calculation of polynomial
coefficients, LSA of an indirect observation with additional constraints was used with Equation 8. The
24 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
new polynomial coefficients approximate the original graticule, but the width and height are the same
as for the original projection.
4.5 Improving the smoothness of the rounded corners
The distinguishing marks of the Natural Earth projection are the rounded corners where bounding
meridians meet the pole lines. By decreasing the length of the pole lines by a small amount, the cured
corners were added during the design process of the original projection, in order to serve five purposes
(Jenny et al. 2007):
(1) The graticule appears more like a spherical Earth;
(2) With a shorter pole line, areal distortion at poles is reduced;
(3) Converging meridians are suggesting that the poles are in fact points;
(4) Curves convey a sense of classic elegance and bring a special esthetic look;
(5) Rounded corners are rare among pseudo-cylindrical projections – they are useful to
distinguish between similar projections.
Two additional measures were required to increase the smoothness of the rounded corners between the
meridians and the pole lines. For the function of coordinate, an additional constraint was added to
the method of least squares, fixing the slope of the polynomial to 7 degrees at the poles. The second
measure for improving the smoothness of the corners was taken by slightly reducing the length of the
pole line from 0.563 to 0.550 before computing the polynomial coefficients. This change reduces a
bulge, which was caused by fixing the slope. The result is a new polynomial Natural Earth projection
that deliberately deviates from the original projection near the poles.
This section continues by providing further details on these two additional measures. Firstly, the
implementation of this additional constraint in matrices of LSA is presented. Secondly, it is described
how the corners can be changed. Thirdly, the usage of the chain rule and graphical explanation is
given. Finally, the value of the slope is explained.
The constraint for fixing the slope of the polynomial to 7 degrees at the poles was added to the method
of least squares. The slope of the polynomial, which expresses the distance of the parallels from the
equator, has a large influence on the slope of meridians at the pole lines. For this reason a derivative of
the coordinate function is used resulting in Equation 22, which adds one row to Equation 21:
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 25
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
(Equation 22)
To change the corners, the slope of meridians must be minimized at 90 degress. Each meridian is a
graph of two variables and . A meridian is a function of the coordinate, where is a parameter:
. When the curve approaches the pole line, the slope decreases, and at the end the final slope
should equal zero (Equation 23) to achieve completely smooth corners. However, the meridians are
not expressed with this function, but each and coordinate is expressed by spherical coordinates as
a parametric curve.
To solve this issue, the chain rule was used to find out which polynomial has the main influence on the
graticule. The derivative of the function with respect to its only parameter (latitude ) was extended
by the chain rule in Equation 23. Using this formula for computing the derivatives, which are
composed of two or more functions, the connection among all applicable derivatives is made. The
partial derivative of coordinate with respect to the longitude is neglected here and the longitude is
used as a constant value ().
(Equation 23)
From Equation 23 it can be seen that the slope of the meridians at 90 degrees (when it is equal to zero)
directly changes the slope of function. It is obvious that the slope of the polynomial which defines
the -coordinate should equal zero (close to zero) to achieve completely smooth corners. In this case,
the slope of the polynomial, which expresses the distance of the parallels from the equator, has the
main influence on the slope of meridians at pole lines. The described connection between these two
slopes is also indicated by the graphs of both tabular parameters (Figure 9).
The graph showing the length of the parallels (Figure 9, left) at 90 degrees is decreasing and the slope
of this curve is the largest at this point. On the other hand, the slope of the second tabular parameter
(Figure 9, right) is slowly decreasing towards 90 degress. Therefore, a change of the slope to the first
curve has a larger effect on the graticule than a change to the slope of the function.
26 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Figure 9: Screenshots of graphs of both tabular parameters with dependence of the latitude in Flex Projector.
Abscissa has latitude values in degrees units, and ordinate contains the values of parameters (without units).
For perfect smoothness at the corners, the slope of the border meridians at pole lines should equal
zero, but this additional constraint adds undesired bulges at the corners. To prevent this deviation, the
slope at the poles from vector in Equation 22 was increased to 7 degrees. With this small slope,
corners are almost invisible, and bulges are reduced. To hide the small bulges in the corners, the length
of the pole line was reduced. This approach does not provide perfect smoothness, but from a
cartographer’s point of view, the corners are not noticeable.
4.6 Inverting the polynomial equation
The inverse of a map projection transforms Cartesian coordinates into spherical coordinates. To
determine the inverse of the polynomial Natural Earth projection, Equation 17 must be inverted, but
analytical expressions of the inverse of polynomials do not exist. However, there are many methods
for solving polynomial systems (Elkadi et al., 2005). For inverting the Natural Earth projection the
Newton-Raphson root finding algorithm was chosen because it converges rapidly, it is easy to
compute, and it requires only one initial guess.
The system defined by the two polynomials (Equation 17) has two known variables (the Cartesian
coordinates and ) and two unknown ones (the spherical coordinates and ). It is solved by
finding the latitude in function and then solving equation for the unknown longitude .
The function from Equation 9 is formed by converting Equation 17 to Equation 24:
(Equation 24)
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 27
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
The iterative approximation is repeated until a sufficiently accurate solution is reached. Convergence
to the solution is quadratic for Equation 24, because the derivative is positive for all
, and has therefore no local minimum or maximum in the valid range of . The
closest local extremum is at , which is outside the valid range of . The quotient
can be used as an initial guess for the Newton-Raphson algorithm, as it is in the range of the latitude
, and does not have any local extremum in this range (Equation 25).
(Equation 25)
An alternative general method for inverting arbitrary map projections without explicit inverse
expressions was described by Ipbüker and Bildirici (2002). They utilize the two forward expressions
to calculate the geographical coordinates and using Jacobian matrices. For the Natural Earth
projection, this method based on Jacobian matrices results in the same values as the Newton-Raphson
approach presented here. For both methods, an equal number of iterations are required (with an
identical ). However, the Newton-Raphson method is faster as it involves fewer calculations, and it is
algorithmically simpler.
28 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
5 ANALYTICAL EXPRESSION FOR THE NATURAL EARTH PROJECTION – RESULTS
The final result of this thesis is the forward polynomial expression of the Natural Earth projection and
its inverse procedure. The equations improve the rounded corners of the graticule, which was also a
goal of this study. In this chapter the forward polynomial expression with all polynomial coefficients
is presented. It is the result of the first five phases from the previous section. The sixth phase provides
the inverse procedure using the Newton-Raphson method, which is explained in the second part of this
section. The last section discusses the difference between the original and the polynomial Natural
Earth projections.
5.1 Forward polynomial expressions
The polynomial expression for the Natural Earth projection was already given in Equations 17 in
section 4.3. The polynomials are of higher degrees than those by Canters and Decleir (1989) for the
Robinson projection, as explained in Chapter 2.4. Higher degrees are required for the Natural Earth
projection to smoothly model the curved corners connecting the meridian lines and the horizontal pole
line.
Equation 26 replaces both and the factor in Equation 1 with a polynomial expression (where
). The polynomial in Equation 27 includes the constant factors , , , and for reducing the
number of required multiplications and accelerating calculations. In order to further accelerate
computations, the number of polynomial terms in the least squares models has been reduced. Equation
26 has no terms with degrees six and eight, and Equation 27 has no term with degree five. Due to the
characteristics presented in Chapter 4.1, Equation 26 contains only even powers of that are
multiplied by longitude , and Equation 27 only consists of odd power terms of .
(Equation 26)
(Equation 27)
where:
and are the projected coordinates,
and are the latitude and longitude in radians,
is the radius of the generating globe, and
to and to are coefficients given in Table 3.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 29
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Table 3: Coefficients for the polynomial expression of the Natural Earth projection.
Coefficients for Equation 26 Coefficients for Equation 27
0.870700
1.007226
-0.131979
0.015085
-0.013791
-0.044475
0.003971
0.028874
-0.001529
-0.005916
The final polynomial coefficients are given in Table 3. They were determined with the Matlab code in
Appendix A written for this purpose. Calculations in Matlab are made with a precision of 10-15. To
simplify the implementation, the parameters given in Table 3 are rounded to six decimal places.
Rounding the parameters results in deviations smaller than 0.02 mm at a scale of 1 : 5,000,000.
Parameters with only five decimal places would result in larger deviations (close to 2 mm) at the same
scale.
Figure 10: The graticule with coastlines computed with the polynomial Natural Earth projection.
5.2 Inverse projection
The inverse projection of the polynomial Natural Earth projection consists of the following steps:
(1) The initial guess for the unknown latitude:
(2) With the Newton-Raphson root finding algorithm, an improved latitude is calculated:
, where is the function from Equation 24, its
derivative, and . At step the iteration stops if , where is
a sufficiently small positive quantity, typically close to the maximum precision of floating
point arithmetic.
30 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
(3) The final latitude:
(4) The final longitude:
The Newton-Raphson method is only applied to compute the latitude in step (2); the longitude can
be computed in step 4 by inverting Equation 26. The Newton-Raphson method converges quickly with
Equation 24. On average, less than four iterations are needed when transforming a regularly spaced
graticule with 15 degrees resolution covering the whole sphere (with ).
5.3 Comparison of the original and the polynomial Natural Earth projection
Since LSA is used for computing a polynomial equation, the two graticules are not identical. There are
some deviations at corners where the lateral meridians meet the pole lines, which is due to the
additional smoothness which was added to this projection.
Table 4: Comparison of the lengths and distances of parallels.
Latitude The Lengths of Parallels The Distances of Parallels from Equator
Original New values Difference Original New values Difference
0 1 1 0.0000 0 0 0.0000
5 0.9988 0.9988 0.0000 0.062 0.0618 -0.0002
10 0.9953 0.9954 0.0001 0.124 0.1236 -0.0004
15 0.9894 0.9895 0.0001 0.186 0.1856 -0.0004
20 0.9811 0.9813 0.0002 0.248 0.2476 -0.0004
25 0.9703 0.9706 0.0003 0.31 0.3098 -0.0002
30 0.957 0.9573 0.0003 0.372 0.372 0.0000
35 0.9409 0.9413 0.0004 0.434 0.4342 0.0002
40 0.9222 0.9225 0.0003 0.4958 0.4962 0.0004
45 0.9006 0.9008 0.0002 0.5571 0.5575 0.0004
50 0.8763 0.8762 -0.0001 0.6176 0.618 0.0004
55 0.8492 0.8488 -0.0004 0.6769 0.677 0.0001
60 0.8196 0.8189 -0.0007 0.7346 0.7344 -0.0002
65 0.7874 0.7868 -0.0006 0.7903 0.7897 -0.0006
70 0.7525 0.7528 0.0003 0.8435 0.8429 -0.0006
75 0.716 0.7167 0.0007 0.8936 0.8934 -0.0002
80 0.6754 0.6763 0.0009 0.9394 0.94 0.0006
85 0.627 0.6256 -0.0014 0.9761 0.9786 0.0025
90 0.563 0.5504 -0.0126 1 1 0.0000
Table 4 presents the differences between the original tabular parameters and the parameters calculated
from the polynomial coefficients in Table 3 by using the Equations 1, 26 and 27. At 0 degree of the
length and at 90 degree of the distance, the results of additional constraints can be seen. Both values
are unchanged, as expected. The value of the distance at zero degree is zero for the original and the
approximated curves. This characteristic did not require an additional constraint in LSA. Since the
polynomial does not contain a constant term in Equation 27, and all terms are multiplied by the
latitude, at 0 degree the value of the parameter is zero. At higher latitudes (85 and 90 degrees) there
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 31
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
are the largest differences, mainly caused by the improvement of the smoothness of the rounded
corners. The biggest deviation of the length at 90 degrees is mainly caused by the manual reduction of
the length of the pole line.
These conclusions can be made referring to Table 4. An objective evaluation of the polinomial
graticule is difficult because the original tabular parameters do not have any units, and the graphical
implications of the deviations are difficult to estimate. The evaluation was made by applying the
graticule at a scale of 1 : 5,000,000, which shows the map of 3.489 m in width and 1.814 m in height.
At this scale a difference in the length of parallels of 0.0001 is reflected along the border-meridian
() by a deviation of 0.35 mm in the coordinate. Horizontal deviations are linearly
decreasing to zero towards the central meridian (). For the distance of parallels, a difference of
0.0001 is equivalent to 0.18 mm of deviation along the coordinate.
The original Natural Earth projection was defined by 19 control points, which were used for the
approximation. Figure 11 presents the absolute deviation from these points in the range of all positive
latitudes. The graph shows large deviations at higher latitude, especially for the latitude at 90 degrees,
which is almost 45 mm, caused by the described improvement of the rounded corners. Otherwise, the
deviations for this graticule are less than 2.5 mm, mainly around the value of 1 mm at scale
1 : 5,000,000. At 60 and 65 degrees deviations are larger but they do not exceed the value of 2.5 mm.
Figure 11: Deviation of the control points on the map at the scale 1: 5,000,000.
To evaluate whether the distortion parameter of the polynomial equation differs from the original
projection, the new polynomial expressions were implemented in Flex Projector, and the distortion
parameters were calculated. Java Code for implementing the polynomial equation in Flex Projector is
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Deviation [mm]
Latitude φ[°]
Deviation from the control points on a map at scale 1 : 5 000 000
32 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
presented in Appendix C. Table 5 provides the comparison of the distortion values from the original
and new (improved) projections. As one can notice, the distortion values do not change significantly.
Table 5: Distortions for the Natural Earth projection computed with Flex Projector.
Original projection New projection
Weighted mean error for overall scale distortion 0.25 0.25
Weighted mean error for areal distortion 0.19 0.19
Mean angular deformation index 20.56 20.54
A graphical comparison of both projections in Flex Projector shows the close resemblance of both
projection graticules. The corner where the pole line meets the meridians can be seen in the magnified
original Natural Earth projection in Figure 12 (A). Picture B shows the smoothed corner of the
polynomial projection. Figure 12 (C) illustrates the reduced pole line. Figure 12 (C) reveals that the
largest deviations are at higher latitudes. The corner is smooth, but a small deviation from the original
curve is present between 80 and 85 degrees.
Figure 12: The original (A) and polynomial (B) Natural Earth projections are overlaid in (C). Arrows indicate
changes in smoothness at the end of the pole line, which is shortened in (B).
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 33
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
6 CONCLUSION
This thesis had two goals. The first goal was to develop an equation for the Natural Earth projection
with an inverse that it is easy to compute and implement in cartographic software. To achieve this
goal, a polynomial expression was chosen to approximate this relatively new projection. Three basic
characteristics were taken into account: the symmetry of the projection, the straight parallels, and the
equally spaced meridians. According to those characteristics, Equations 26 and 27 were derived. For a
good approximation of the original graticule, the order of the polynomials equals 13 and 11. The
polynomial coefficients were defined by the LSA approximation method with additional constraints to
preserve the width and height of the projection graticule. The reference variance and residuals from
LSA were used to evaluate the differences between the approximated and the original projection. The
equations are easy to code and fast to compute as only seven multiplications are required for each
polynomial if factorized appropriately. The proposed inverse algorithm uses the Newton-Raphson
method and requires only four iteration steps. It is slightly slower in comparison to an analytical
inverse formula, but it converges quickly. On average, less than four iteration steps are required.
The Natural Earth projection expressed by the polynomial Equations 26 and 27 slightly deviates from
Patterson’s original projection by adding additional curvature to meridians where they meet the
horizontal pole line. The curved corners are smoother than in the original design, which improves the
visual appearance of the graticule. The polynomial equations were developed in collaboration with
Tom Patterson and during this development he expressed his desire to smooth the edges at the ends of
the pole lines. This second goal of this thesis was reached by changing the polynomial coefficients. In
the functional model used for LSA, an additional constraint was added – fixing the slope of the
coordinate at 7 degrees. Considering this constraint and the manual reduction of the pole line, the
slope of meridians at the pole line becomes so small that the edges are no longer visible.
Since the scale distortion index, the areal distortion index, as well as the mean angular deformation
index (Canters and Decleir, 1989) of the polynomial approximation of the Natural Earth projection are
almost identical to those of the original projection, the authors of the paper “A polynomial equation of
the Natural Earth projection” (Šavrič et al., accepted for publication) recommend using the polynomial
equation (Equations 26 and 27 in Table 3) instead of the original Natural Earth projection. Details on
LSA with constraints for obtaining the polynomial formulas are also provided in this paper to allow
others to apply this technique to similar projections defined by tabular parameters. It remains to be
explored how the polynomial approximation method can be generalized for any projection designed
with Flex Projector.
34 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Figure 13: Screenshot of the Flex Projector with the improved Natural Earth projection.
The improved Natural Earth projection is implemented in the last version of the software application
Flex Projector 0.118, released on 6 April 2011 (Jenny, B., Patterson, T. 2007). Figure 13 shows the
screenshot of the Flex Projector with the Natural Earth projection.
Due to the fact that the Natural Earth projection now has its own polynomial equation, one can hope
that the publication of the formulas will help the projection find its place in other cartographic
projection libraries and software applications. Global Mapper is the first commercial software to
include the developed polynomial Natural Earth projection. Developers of GeoCart, Natural Scene
Designer, Java Map Projection Library (Appendix C), PROJ.4, MAPublisher, Geographic Imager and
others have added the new Natural Earth projection to their software, which will be released in future
updates. Figure 14 shows a screenshot of Global Mapper version 12.02.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 35
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Figure 14: Screenshot of Global Mapper with the polynomial Natural Earth projection (Global Mapper, 2009).
36 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
7 RAZŠIRJEN POVZETEK DIPLOMSKE NALOGE V SLOVENSKEM JEZIKU
P.1 UVOD
P.1.1 Opis problema
V zgodovini so se kartografi pogosto soočali s težavami pri oblikovanju novih kartografskih projekcij.
Pomanjkanje matematičnega znanja in zamudno prikazovanje rezultatov sta bila glavni težavi razvoja.
Leta 1974 je Arthur H. Robinson predstavil nov način določanja kartografske projekcije, ki ga je
uporabil za razvoj svoje, danes svetovno znane in uporabljane Robinsonove projekcije. Njegov pristop
je povsem grafičen in čeprav je s tega vidika enostavnejši, ne omogoča eksaktne določitve analitične
enačbe, ki povezuje sferne in projekcijske koordinate.
Tom Patterson je leta 2007 zasnoval Naravno Zemljino projekcijo z uporabo programa Flex Projector
(Jenny et al., 2008). Ker program uporablja grafični pristop Robinsona, Naravna Zemljina projekcija
ni določena z analitično enačbo. Zasnovana je na podlagi dveh parametrov – dolžine paralel in njihove
oddaljenosti od ekvatorja. Oba parametra sta podana za vsakih pet stopinj geografske širine in kot taka
definirata pare kontrolnih točk. Krivulje meridianov skozi te točke določa interpolacijska metoda
kubičnih zlepkov. Ta metoda interpolacije je enostavna za preračun, vendar vsebuje veliko število
parametrov in zahteva kar precej programiranja, če želimo projekcijo uporabiti v drugih programskih
orodjih. Tako uporaba te projekcije ostaja omejena na program, v katerem je bila zasnovana.
P.1.2 Glavni cilj naloge
Z namenom, da bi razširili uporabo Naravne Zemljine projekcije tudi na ostale programske pakete, je
glavni cilj te diplomske naloge določitev analitične enačbe projekcije. Analitična enačba mora
omogočati transformacijo sferičnih v katarzične koordinate za celotno definicijsko območje. Pretvorba
mora biti enostavna, s čim manj parametri in imeti določeno tudi inverzno funkcijo. Programiranje naj
bo enostavno in hitro.
P.1.3 Dodatni cilj naloge
Posebnost Naravne Zemljine projekcije je stičišče robnega meridiana in pola, ki je v tej projekciji
predstavljen kot daljica. Ob večji povečavi se na tem stiku pojavi nezaželeni lom (Slika P. 1).
Snovalec projekcije v programu Flex Projector loma ni uspel zgladiti, zato je dodatni cilj te naloge
izboljšati gladkost omenjenega stika.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 37
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Slika P. 1: Povečava stičišča projekcij meridiana in pola. Opazen lom je označen z rdečim krogom.
P.2 ZASNOVA IN IZVOR NARAVNE ZEMLJINE PROJEKCIJE
P.2.1 Naravna Zemljina kartografska projekcija
Naravno Zemljino projekcijo je zasnoval Tom Patterson leta 2007 kot poskus izdelave projekcije, ki bi
bolje kot vse dosedanje zadovoljila pogoje za prikaz karte celotnega fizičnega površja Zemlje v
majhnem merilu (Jenny et al., 2008). Za sam nastanek projekcije je bilo uporabljeno iterativno orodje
za grafično oblikovanje kartografskih projekcij, imenovano Flex Projector. Projekcija je kombinacija
dveh že obstoječih kartografskih projekcij –
Kavraiskiy VII in Robinsonove (Jenny et al.,
2008).
Obliko mreže meridianov in paralel določata dva
parametra: (1) dolžina paralel () in (2)
oddaljenost paralel od ekvatorja (). Vrednosti
obeh parametrov so za Naravno Zemljino
projekcijo podane na vsakih pet stopinj geografske
širine in so brez enot (Tabela P. 1). Parametri so
bili določeni z iterativnimi poizkusi in na podlagi
vrednotenja podob kontinentov na sami projekciji
(Jenny et al. 2008). Njihovo pretvorbo v
projekcijske koordinate podajata spodnji enačbi:
Tabela P. 1: Vrednost parametrov za Naravno
Zemljino projekcijo (Jenny, et al., 2008, stran 25).
Geografska
širina
Dolžina
paralel
Oddaljenost
paralel od
ekvatorja
0 1 0
5 0.9988 0.062
10 0.9953 0.124
15 0.9894 0.186
20 0.9811 0.248
25 0.9703 0.31
30 0.957 0.372
35 0.9409 0.434
40 0.9222 0.4958
45 0.9006 0.5571
50 0.8763 0.6176
55 0.8492 0.6769
60 0.8196 0.7346
65 0.7874 0.7903
70 0.7525 0.8435
75 0.716 0.8936
80 0.6754 0.9394
85 0.627 0.9761
90 0.563 1
38 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
V Enačbi P. 1 sta in projekcijski ter in sferni koordinati. R je polmer Zemljine krogle,
medtem ko predstavlja linijsko merilo na ekvatorju in določa lego sekantnega valja z
dvema standardnima paralelama, pa je razmerje med dolžino srednjega meridiana in
ekvatorja.
Oba parametra in podajata pare kontrolnih točk skozi katere potekajo krivulje meridianov.
Vmesne točke so pri osnovni definiciji podane z interpolacijsko metodo kubičnih zlepkov, ki jo za ta
namen uporablja tudi Flex Projector. Tabela P. 1 in Enačba P. 1 skupaj z interpolacijsko metodo tako
predstavljata transformacijo med sferičnimi in projekcijskimi koordinatami, ki jo želimo v tej
diplomski nalogi poenostaviti.
Slika P. 2: Mreža meridianov in paralel Naravne Zemljine projekcije za vsakih 30° geografske širine in dolžine.
Z rdečo barvo sta prikazani tudi koordinatni osi.
Kot je razvidno iz zgornje slike, je Naravna Zemljina projekcija psevdocilindrična projekcija z
enakomerno razporeditvijo meridianov ter ravnimi paralelami. Osi in predstavljata projekcijo
ekvatorja in srednjega meridiana (Slika P. 2). Mreža je simetrična glede na obe koordinatni osi.
Dolžina ekvatorja je zmanjšana za merilo , medtem ko se srednji meridian preslika kot ravna
premica, in je 0,52-kratnik dolžine ekvatorja. Ostali meridiani so s svojo konkavno stranjo obrnjeni
proti srednjemu meridianu. Stičišča robnih meridianov in daljic polov so zakrivljena, kar je
prepoznavna lastnost te projekcije. Zakrivljenost ostalih meridianov se zmanjšuje takrat, ko se od roba
približujemo srednjemu meridianu, kateri pravokotno seka daljici polov.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 39
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Naravna Zemljina projekcija ni niti konformna niti ekvivalentna. Njene distorzijske vrednosti so
primerljive z ostalimi projekcijami. Kot pri vseh podobnih psevdocilindričnih projekcijah, so največje
deformacije prisotne na polih (Jenny et al., 2008). Slike 3, 4 in 5 na straneh 7 in 8 v osnovnem delu te
diplomske naloge prikazujejo razporeditev vrednosti deformacij.
P.2.2 Analitične enačbe za Robinsonovo projekcijo
Naravna Zemljina projekcija zagotovo ni prva takšna projekcija, ki nima osnovne analitične enačbe.
Prvo takšno projekcijo je ustvaril sam snovalec grafičnega pristopa, Arthur H. Robinson. Njegova
Robinsonova projekcija je bila že od vsega začetka predmet debat, v katerih so se spraševali kako
uspešno modelirati projekcijo skozi pare kontrolnih točk. V svojem bistvu se ločita dva načina
modeliranja – aproksimacija in interpolacija. Interpolacijska metoda kubičnih zlepkov je na primeru
Naravne Zemljine projekcije že uporabljena in tudi neprimerna zaradi velikega števila parametrov ter
oteženega programiranja. Veliko enostavnejše so aproksimacijske metode, pa čeprav te ne podajo
krivulje, ki bi potekale natančno skozi kontrolne točke.
Za Robinsonovo projekcijo sta bili razviti dve aproksimaciji. Prvo sta predstavila Canters in Decleir
(1989). Gre za aproksimacijo z dvema polinomskima enačbama pete stopnje, ki skupno vsebujeta le
šest parametrov. Drugo rešitev aproksimacije je dve leti za tem predstavil Beineke (1991, 1995). Za
koordinato je uporabil polinomsko enačbo sedme stopnje, medtem ko je koordinato izrazil z
eksponentno enačbo. Čeprav je druga rešitev nekoliko boljša, je izračun prve enostavnejši in hitrejši.
To dokazuje tudi manjša simulacija v Java programskem jeziku. Ob enakem številu parametrov je
eksponentna enačba nekaj več kot desetkrat počasnejša za preračun od polinomske. Edina slabost
polinomskih enačb je ta, da zanje pogosto ne obstaja inverzna funkcija. Ta težava se v večini primerov
rešuje z uporabo drugih numeričnih metod reševanja, kot sta recimo tangentna (Newtonova) metoda in
metoda bisekcije. Glede na vsa zgoraj navedena dejstva smo se odločili za izbiro polinomske oblike
enačbe za določitev Naravne Zemljine projekcije.
P.3 UPORABLJENE NUMERIČNE METODE
P.3.1 Metoda najmanjših kvadratov – posredna izravnava
V prvih fazah razvoja enačb je bila uporabljena posredna izravnava po metodi najmanjših kvadratov.
Število enačb v tem matematičnem modelu je enako številu podanih vrednosti za vsak parameter ().
Vrednost parametra se obravnava kakor opazovanje, polinomski koeficienti pa nastopajo kot neznanke
(). Enačba P. 2 predstavlja matrični zapis, ki se ga rešuje po Enačbi P. 3 (Mikhail, Ackerman, 1976).
40 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
, (Enačba P. 2)
, (Enačba P. 3)
,
Vektor dimenzije je vektor opazovanj, dimenzije je vektor neznank, je matrika
koeficientov z rangom , vektor predstavlja vektor odstopanj in je vektor konstant, ki
nastopajo v enačbah opazovanj. Pri izračunu si pomagamo z matriko koeficientov normalnih enačb
dimenzije ter z vektorjem stalnih členov (). Matrika je matrika uteži med opazovanji
(Mikhail, Ackerman, 1976).
Ker so vse vrednosti parametrov izražene s polinomi, gre v tem konkretnem primeru za linearni sistem
enačb, ki za rešitev ne potrebuje približnih vrednosti neznank in iterativnega procesa reševanja.
Matrika uteži je enaka enotski matriki, saj so posamezne vrednosti parametrov neodvisne druga od
druge in določene na enak način. Za medsebojno primerjavo rezultatov aproksimacij po tej metodi
uporabljamo vrednost največjega popravka iz vektorja in pa referenčno varianco, določeno po
Enačbi P. 4, kjer predstavlja nadštevilnost.
(Enačba P. 4)
P.3.2 Posredna izravnava s funkcijsko odvisnimi neznankami
Ta, nekoliko dograjen model posredne izravnave, je bil uporabljen za končno določitev vrednosti
polinomskih koeficientov. Enačbam opazovanj smo dodali nov sistem t.i. veznih enačb (), ki opisuje
medsebojno odvisnost neznank (Enačba P. 5). Matrika je matrika koeficientov, pa vektor
konstant. S tem povečamo tudi nadštevilnost enačb, saj velja povezava: . Ob tem
zagotovimo izbrane medsebojne pogoje, ki jih želimo doseči ali pa aproksimacijsko krivuljo položimo
natančno skozi izbrano točko. Oba sistema enačb (Enačbi P. 2 in 5) rešimo po Enačbi P. 6
(Mikhail, Ackerman, 1976).
(Enačba P. 5)
(Enačba P. 6)
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 41
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
P.3.3 Tangentna (Newtonova) metoda reševanja enačb
Tangentna metoda je numerična metoda reševanja enačb oblike . Temelji na iterativnem
načinu reševanja z le enim začetnim približkom rešitve. Pogosto se uporablja za reševanje
polinomskih enačb višjih redov, kadar zanje ne obstaja inverzna funkcija. Če je približek dovolj blizu
pravi rešitvi in je naklon krivulje na tem območju enakega predznaka, iteracija kvadratično konvergira
proti končni rešitvi (Bronštejn et al. 1974). Ta metoda je v našem primeru uporabljena za inverzno
transformacijo iz kartezičnih v sferne koordinate. Opisuje jo Enačba P. 7, kjer je podana
enačba in njen prvi odvod. in predstavljata začetno in novo rešitev enačbe, n in
pa sta iteracijska koraka.
, (Enačba P. 7)
V geometrijskem pomenu, metoda ne išče presečišča funkcije z abscisno osjo, temveč presečišče
tangente na krivuljo v začetni točki s to osjo. Od tu tudi izhaja ime metode (Bronštejn et al. 1974).
P.4 APROKSIMACIJA NARAVNE ZEMLJINE PROJEKCIJE S POLINOMSKO ENAČBO
Pri določitvi analitične enačbe za Naravno Zemljino projekcijo je bila uporabljena aproksimacija s
polinomsko enačbo. Pri tem smo upoštevali dva kriterija, in sicer kot prvi majhno število členov v
enačbi, kot drugi pa absolutno razliko med originalno in nadomestno polinomsko določeno projekcijo,
ki naj bo čim manjša. Prvi kriterij zagotavlja poenostavitev in hitrejši izračun, drugi pa rezultat, ki
odraža podobnost.
Pri razvoju enačbe je sodeloval snovalec projekcije, Tom Patterson. Ker je cilj naloge tudi izboljšava
zakrivljenih robov, ima pri nalogi njegova grafična ocena veliko večji pomen kot absolutna razlika
med originalno in nadomestno, s polinomsko enačbo določeno projekcijo (v nadaljevanju uporabljamo
izraz polinomska Naravna Zemljina projekcija). Celoten proces določitve vsebuje šest ločenih faz, ki
so predstavljene v naslednjih podpoglavjih.
P.4.1 Oblikovanje polinomskih enačb
Obe projekcijski koordinati in sta izraženi z dvema sfernima, in . Tako splošna polinomska
enačba (Enačba P. 8) za primer tretje stopnje vsebuje 20 členov za obe projekcijski koordinati. Vendar
pa moramo splošno obliko nekoliko preoblikovati, saj so zaradi lastnosti psevdocilindrične projekcije
42 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
vrednosti nekateri polinomskih koeficientov blizu ničle in je njihova uporaba tako nesmiselna. Pri
oblikovanju obeh polinomskih enačb smo zato upoštevali naslednje tri lastnosti Naravne Zemljine
projekcije (Canters, 2002):
(1) paralele so ravne, a ne enakomerno porazdeljene daljice,
(2) meridiani so enakomerno porazdeljeni in
(3) projekcija je simetrična glede na obe koordinatni osi.
(Enačba P. 8)
Iz prve lastnosti izhaja, da je koordinata odvisna le od geografske širine , zato iz nje odstranimo
vse člene, ki vsebujejo geografsko dolžino. Druga lastnost spreminja koordinato. Zato, ker so
meridiani enakomerno porazdeljeni, ta enačba zato vsebuje le linearne člene geografske dolžine. Na
koncu pa imamo še simetrijo. Zaradi simetrije skozi obe osi, koordinata vsebuje le člene s sodo,
koordinata pa z liho potenco geografske širine (Canters, 2002). Rezultat za tretjo stopnjo polinoma
prikazuje Enačba P. 9.
P.4.2 Povečanje polinomske stopnje
Tretja stopnja polinoma nikakor ne zadošča za uspešno aproksimacijo, saj želimo zmanjšati tudi
absolutno razliko med projekcijsko mrežo izvorne in polinomske Naravne Zemljine projekcije. Pri
nižjih stopnjah polinoma je aproksimacijska krivulja precej različna od izvorne. Po drugi strani pa
prevelika stopnja prinese krivuljo, ki valovi med kontrolnimi točkami, kar pa tudi ni želen rezultat. S
postopnim dvigovanjem stopnje polinoma smo tako iskali tisti polinom, pri katerem so razlike še
relativno majhne in je krivulja najbolj podobna originalni.
Za vsako koordinato posebej smo ločeno izvedli posredno izravnavo z različnimi stopnjami
polinomov. Za primerjavo rezultatov so bile uporabljene vrednosti največjih popravkov posamezne
izravnave in referenčna varianca. Ko se z večjo stopnjo polinoma vrednosti teh količin niso več
bistveno spremenile, so bile enačbe vpeljane v program Flex Projector. Ta je omogočil grafični izris
koordinatne mreže in s tem vizualno primerjavo izvorne in polinomske koordinatne mreže. Tabela P. 2
in Slika 8 na strani 20 prikazujeta primerjavo različnih stopenj. Na Sliki P. 3 vidimo grafično
primerjavo zadnjih treh redov polinoma, kjer grafične razlike med različnimi redi niso več opazne.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 43
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Za Naravno Zemljino projekcijo je bila po grafični oceni izbrana dvanajsta stopnja polinoma (samo pri
geografski širini ).
Slika P. 3: Primerjava izvirne in polinomskih Naravnih Zemljinih projekcij. Pri uporabljenih polinomih 10., 12.,
in 14. stopnje razlike niso več opazne.
Tabela P. 2: Referenčne variance in največji popravki za različne polinomske stopnje po izravnavi.
Zadnja vrstica prikazuje nove vrednosti po odstranitvi členov z majhnim prispevkom za 12. stopnjo.
Opomba: Enote so mm na karti v merilu 1 : 5.000.000, kjer je polmer Zemlje enak 6.378.137 metrov.
Polinomska
stopnja
Dolžina paralel (
) Oddaljenost paralel (
)
[mm] max (v) [mm]
[mm] max (v) [mm]
6. 7,322 14,701 2,564 5,483
8. 2,328 4,556 1,323 2,447
10. 0,494 1,186 0,499 0,760
12. 0,406 1,173 0,291 0,813
14. 0,359 0,911 0,186 0,476
*12. 0,406 1,081 0,287 0,823
P.4.3 Odstranitev vmesnih členov in končna oblika enačb
Z dvigovanjem polinomske stopnje zmanjšamo absolutno razliko, a s tem povečamo število členov v
enačbah. Nekateri vmesni členi prispevajo h končnemu rezultatu (vrednosti koordinate) manj kakor
ostali. Z namenom poenostavitve izračuna in zmanjšanja števila členov v enačbi so bili v tej fazi
odstranjeni posamezni vmesni polinomski členi.
0.5500
0.6000
0.6500
0.7000
0.7500
0.8000
0.8500
60 65 70 75 80 85 90
Dolžina paralel lφ
Geografska širina φ[°]
Primerjava izvirne in polinomskih (različni red) Naravnih Zemljinih projekcij
Naravna Zemljina projekcija polinom 12. reda
polinom 14. reda polinom 10. reda
44 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Majhne vrednosti koeficientov po izravnavi s polinomom dvanajste stopnje nakazujejo možnost
odstranitve šeste in osme potence geografske širine za koordinato. Polinomu, ki izraža koordinato
pa smo odstranili peto potenco. Po ponovni izravnavi z novimi polinomi se projekcijska mreža ni
bistveno spremenila (Tabela P. 2, vrednosti za stopnjo *12.). Vrednosti referenčne variance za oba
primera sta se celo izboljšala. Enačba P. 10 prikazuje končno obliko enačb, kjer oba polinoma
nadomeščata parametra in ter konstante , in .
P.4.4 Dodajanje pogojev v izravnavo
Uporaba aproksimacijske metode in posredne izravnave v nobenem primeru ne prinaša enačbe, ki bi
podala popolnoma enake dimenzije in obliko projekcijske mreže. Na prav vsaki kontrolni točki se
polinomska mreža razlikuje od originalne za vrednost popravka po izravnavi. Kljub vsemu želimo, da
bi ohranili vsaj enake dimenzije projekcije in s tem vsaj delno zmanjšali razlike med projekcijama. Za
dosego tega smo v posredno izravnavo vpeljali dva dodatna pogoja, torej, da se dolžini ekvatorja in
srednjega meridiana morata ohraniti. Z drugimi besedami to pomeni, da se dolžina paralele pri 0° in
oddaljenost paralele pri 90° geografske širine ohranjata.
S kombinacijo Enačb P. 1 in 10 oba pogoja zapišemo v obliki:
(Enačba P. 11)
Ker v izravnavi uporabljamo linearne enačbe, oba pogoja izrazimo kot medsebojno odvisnost
parametrov. Od te faze naprej se vrednosti polinomskih koeficientov določajo po posredni metodi
izravnave s funkcijsko odvisnimi neznankami (Enačba P. 6).
P.4.5 Zgladitev stičišča robnega meridiana in projekcije pola
Da bi se izboljšala gladkost stičišča robnega meridiana in pola, mora biti krivulja robnega meridiana
oblikovana na način, da se v stičišču z daljico pola njena vrednost naklona ustavi natanko
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 45
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
pri nič stopinjah. To pomeni, da mora biti odvod v tej točki enak nič. Žal pa ne poznamo funkcije v
odvisnosti od koordinate.
Ker imamo za vsako koordinato svojo parametrično funkcijo, moramo v tem primeru ugotoviti, katera
je tista, ki ima največji vpliv na končno vrednost naklona. Tako razširimo parcialni odvod
koordinate z verižnim pravilom, kakor je prikazano v Enačbi P. 12. Geografsko dolžino pri tem
obravnavamo kot konstanto , saj se ta situacija rešuje le za primer robnega meridiana in je
zato ne navajamo v nadaljnjih enačbah.
(Enačba P. 12)
Funkciji obeh koordinat sta medsebojno neodvisni, zato sprememba ene ne vpliva na drugo. Obe pa
vplivata na funkcijo v odvisnosti od koordinate. Podobno velja tudi za odvode, ki so predstavljeni
v Enačbi P. 12. Iz te enačbe tako sledi, da sprememba odvoda koordinate po geografski širini
povzroči spremembo vrednosti odvoda
, v našem primeru je to vrednost naklona robnega meridiana.
Pri tem odvod koordinate po geografski širini
ostaja ves čas enak.
Popolno analitično gladkost robov dosežemo, če je naklon robnega meridiana v točki pola enak
natančno nič. V izravnavo tako dodamo pogoj, ki odvod koordinate v tej točki nastavi na vrednost
nič:
. Po Enačbi P. 12 tako spremenimo tudi naklon robnega meridiana. Vendar pa tak dodaten
pogoj v izravnavi povzroči nezaželeno odstopanje (izboklino) na mreži projekcije. Ta razlika je
precejšnja in posledica tega so spremenjena oblika projekcije in prevelika odstopanja v kontrolnih
točkah. Ker vseeno želimo prikriti lom, smo se odločili za kompromisno rešitev. Naklon grafa smo
najprej povečali na sedem stopinj. Zato, da bi prikrili nastalo izboklino, smo nato še dodatno zmanjšali
dolžino daljice pola. Sedem stopinjski naklon omogoči, da uporabnik loma ne zazna pri veliki, a še
smiselno uporabljeni povečavi. Rezultat je tako boljši kot pri izvorni projekciji, kjer je bilo lom
mogoče dokaj hitro zaznati.
Za grafično gladkost zakrivljenih robov Naravne Zemljine projekcije je bilo tako potrebno dvoje.
Najprej smo v izravnavo dodali tretji pogoj: naklon krivulje koordinate v odvisnosti od geografske
širine na polu (v točki ) je bil določen na 7 stopinj (Enačba P. 13). Ker pa vsiljen naklon
povzroči manjšo izboklino na robnem meridianu je bila dolžina paralele na polu reducirana iz 0.563 na
0.550 že pred samo izravnavo. S tem se manjše odstopanje prikrije in izboklina je potisnjena
navznoter. Rezultat teh dveh dodatnih sprememb je polinomska aproksimacija, ki obenem tudi
46 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
izboljšuje mrežo Naravne Zemljine projekcije in je navedena v poglavju P.5.1 kot rezultat te
diplomske naloge.
(Enačba P. 13)
P.4.6 Inverzna projekcija
Za določitev inverznih enačb projekcije izhajamo iz sistema dveh enačb (Enačba P. 10), kjer
projekcijski koordinati in nastopata kot dani vrednosti ter sferni koordinati in kot neznanki.
Enačba koordinate vsebuje le eno neznanko, , katere vrednost nato uporabimo v drugi enačbi za
izračun neznanke . Princip je v svoji osnovi enostaven, vendar zahteva invertiranje polinomske
enačbe 11. stopnje.
Na splošno ne obstaja analitični inverzni način reševanje polinomskih enačb z realnimi koeficienti.
Takšne sisteme se zato pogosto rešuje z različnimi numeričnimi metodami. Za invertiranje Naravne
Zemljine projekcije smo izbrali tangentno metodo, saj metoda izpolnjuje pogoj konvergence. Odvod
enačbe je na celotnem definicijskem območju
pozitiven in zato tukaj ni nobenega
lokalnega ekstrema. Hkrati je kvocient lahko uporabljen kot začetna vrednost, saj so vse
njegove vrednosti znotraj definicijskega območja geografske širine.
P.5 ANALITIČNA ENAČBA NARAVNE ZEMLJINE PROJEKCIJE – REZULTATI
P.5.1 Polinomski enačbi projekcije
Enačba P. 14 nadomešča tabelarični parameter in merilo ekvatorja iz Enačbe P. 1 s polinomom
(kjer je ). Polinom v Enačbi P. 15 vsebuje parametre , , , in . Integracija teh konstant v
polinome je bila narejena z enim samim namenom, in sicer zato, da se zmanjša število množenj in s
tem omogočiti hitrejši preračun. Enačba P. 14 tako vsebuje člene s sodimi potencami geografske
širine, brez šeste in osme, le enkrat pomnožene z geografsko dolžino. V Enačbi P. 15 so le členi lihe
potence geografske širine, brez pete, in brez geografske dolžine.
(Enačba P. 14)
(Enačba P. 15)
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 47
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
kjer so:
in projekcijske koordinate,
in geografska širina in dolžina, podani v radianih,
je polmer zemlje,
do in do so koeficienti, podani v Tabeli P. 3.
Tabela P. 3: Koeficienti polinomski enačb za Naravno Zemljino projekcijo.
Koeficienti za X Koeficienti za Y
0,870700
1,007226
-0,131979
0,015085
-0,013791
-0,044475
0,003971
0,028874
-0,001529
-0,005916
P.5.2 Inverzna projekcija
Inverzno projekcijo sestavljajo naslednji štirje koraki:
(1) Izračun začetne vrednosti geografske širine:
(2) S tangentno metodo sledi izračun geografske širine: ,
kjer je , njen
odvod in . V -tem koraku se iterativni proces ustavi, ko je izpolnjen pogoj
, kjer je zadovoljivo majhno pozitivno število.
(3) Končna vrednost geografske širine:
(4) Izračun geografske dolžine:
Tangentna metoda je pri tem uporabljena samo enkrat, in sicer v drugem koraku za rešitev
Enačbe P. 15. Metoda konvergira hitro, v povprečju so potrebne manj kot 4 iteracije.
P.5.3 Primerjava izvirne in polinomske Naravne Zemljine projekcije
Primerjava obeh projekcij je bila izvedena na projekcijski mreži v merilu 1 : 5.000.000, kjer je širina
karte 3,489 m in višina 1,814 m. Slika P. 4 predstavlja absolutne razlike 19 kontrolnih točk na robnem
meridianu, , s katerimi je bila originalna projekcija tudi podana. Graf prikazuje večja
odstopanja pri večjih vrednostih geografske širine, še posebej pri polu, kjer razlika znaša skoraj
45 mm. Razlog za to spremembo je seveda izboljšava zakrivljenih robov. Na Sliki P. 5 lahko vidimo,
kako se je ta zakrivljenost spremenila in kakšne razlike so prisotne med mrežama obeh projekcij.
Nekoliko večje odstopanje je pri 60. in 65. stopinjah, v ostalih točkah pa so vrednosti okoli 1 mm.
48 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Slika P. 4: Odstopanja kontrolnih točk robnega meridiana na karti merila 1 : 5.000.000.
Slika P. 5: Izvirna (A) in izboljšana (B) Naravna Zemljina projekcija. Puščice nakazujejo spremembo gladkosti
ob stičišču pola in robnega meridiana.
P.6 ZAKLJUČEK
Glavna naloga diplomske naloge je bila določitev analitične enačbe za Naravno Zemljino projekcijo.
Za dosego cilja je bila uporabljena aproksimacijska metoda s polinomskimi enačbami. Oblikovanje
enačb je temeljilo na osnovnih lastnostih projekcije in iskanju minimalnih absolutnih razlik med
izvorno in izboljšano projekcijo. Predstavljeni polinomski enačbi, ki nadomeščata konstante in
tabelarične vrednosti vsebujeta obvladljivo število parametrov in sta enostavni za preračun ter
integracijo v računalniške programe. Ob pravilni faktorizaciji enačb zahtevata vsaka le sedem
množenj. Projekcijo je mogoče tudi invertirati z uporabo tangentne metode, za katero v povprečju
potrebujemo manj kot štiri korake.
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Odstopanja [mm]
Geografska širina φ[°]
Odstopanja kontrolnih točk robnega meridiana na karti merila 1 : 5 000 000
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 49
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Projekcija izboljšuje zakrivljenost stičišč in grafično podaja videz, da so sedaj le-ta zglajena. Zaradi
tega lahko smatramo, da polinomske enačbe določajo izboljšano Naravno Zemljino kartografsko
projekcijo. Avtorji članka “A polynomial equation of the Natural Earth projection” (Šavrič et al.,
sprejet za objavo) tako predlagajo uporabo polinomskih enačb (Enačbi P. 14 in 15 ter Tabela P. 3) kot
pravo analitično enačbo za Naravno Zemljino projekcijo, ki bi nadomestila dosedanjo grafično
določitev projekcije.
S tem sta izpolnjena tako glavni kakor tudi dodatni cilj te diplomske naloge. Do sedaj Naravno
Zemljino kartografsko projekcijo že vključuje Java Map Projection Library (Dodatek, Apprendix C) in
je seveda v seznamu projekcij, ki so vključene v aplikaciji Flex Projector. Objava polinomskih enačb
pa bo omogočila širšo uporabo te kartografske projekcije tudi v drugih programskih orodjih. Prvi
komercialni program, ki omogoča uporabo te kartografske projekcije, je tako že Global Mapper
(Slika 14, stran 34). Ob izidu novih verzij programov GeoCart, Natural Scene Designer, PROJ.4,
MAPublisher, Geographic Imager in drugih bodo ti prav tako vsebovali Naravno Zemljino
kartografsko projekcijo.
Diplomska naloga predstavlja način določitve polinomske enačbe konkretno na primeru Naravne
Zemljine kartografske projekcije, ki je bila grafično zasnovana v programu Flex Projector. Objava
uporabljene metode v članku (Šavrič et al., sprejet za objavo) bo omogočila uporabo te tehnike tudi za
druge projekcije, ki so bile zasnovane na enak način. Nerešena ostaja le umestitev te polinomsko
aproksimacijske metode v Flex Projektor za katero koli projekcijo v tem programu.
50 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
8 REFERENCES
8.1 Used references
Beineke, D. 1991. Untersuchung zur Robinson-Abbildung und Vorschlag einer analytischen
Abbildungsvorschrift. Kartographische Nachrichten 41, 3: 85–94.
Beineke, D. 1995. Kritik und Diskussion: Zur Robinson-Abbildung. Kartographische Nachrichten 45,
4: 151–3.
Bretterbauer, K. 1994. Ein Berechnungsverfahren für die Robinson-Projektion. Kartographische
Nachrichten 44, 6: 227–9.
Bronštejn, J. N., Semendjajev, K.A. 1974. Matematični priročnik. Ljubljana, Tehniška založba
Slovenije: p. 163-5.
Canters, F. 2002. Small-scale map projection design. London, Taylor & Francis: 336 p.
Canters, F., Decleir, H. 1989. The world in perspective – A directory of world map projections.
Chichester, John Wiley and Sons: p. 143.
Elkadi, M., Mourrain, B. 2005. Symbolic-numeric methods for solving polynomial equations and
applications. In: Dickenstein, A. (eds.), Emiris, I. Z. (eds.). Solving Polynomial Equations:
Foundations, Algorithms and Applications, volume 14 of Algorithms and Computation in
Mathematics. Berlin, Germany: Springer, 125–68.
Evenden, I. E. 2008. libproj4: A comprehensive library of cartographic projection functions
(preliminary draft).
http://home.comcast.net/~gevenden56/proj/manual.pdf (accessed April 12, 2011).
Global Mapper Software LLC, 2009.
http://www.globalmapper.com/ (accessed May 15, 2011)
Ipbüker, C. 2004. Numerical evaluation of the Robinson projection. Cartography and Geographic
Information Science 31, 2: 79–88.
Ipbüker, C. 2005. A computational approach to the Robinson projection. Survey Review 38, 297:
204–17.
Ipbüker, C., Bildirici, I. Ö. 2002. A general algorithm for the inverse transformation of map
projections using Jacobian matrices. Proceedings of the Third International Symposium Mathematical
& Computational Applications: 175–82.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 51
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Jenny, B. 2011. Java Map Projection Library.
http://sourceforge.net/projects/jmapprojlib/ (accessed May 15, 2011)
Jenny, B. and Patterson, T. 2007. Flex Projector.
http://www.flexprojector.com (accessed May 15, 2011).
Jenny, B., Patterson, T., Hurni, L. 2008. Flex Projector – interactive software for designing world map
projections. Cartographic Perspectives 59: 12–27.
Jenny, B., Patterson, T., Hurni, L. 2010. Graphical design of world map projections. International
Journal of Geographical Information Science 24, 11: 1687–702.
Mikhail, E. M., Ackerman, F. 1976. Observations and least squares. New York, Harper & Row,
Publishers: p. 99-174, 213-228.
Ratner, D. A., 1991. An implementation of the Robinson map projection based on cubic splines.
Cartography and Geographic Information Systems 18: 104–8.
Richardson, R.T. 1989. Area deformation on the Robinson projection. The American Cartographer 16:
294–6.
Robinson, A. 1974. A new map projection: its development and characteristics. In: Kirschbaum, G. M.
(eds.), Meine, K.-H. (eds.). International Yearbook of Cartography, Bonn-Bad Godesberg, Germany:
Kirschbaum, 145–55.
Snyder, J.P. 1990. The Robinson projection: a computation algorithm. Cartography and Geographic
Information Systems 17: 301–5.
Snyder, J. P. 1993. Flattening the earth. Two thousand years of map projections, University of
Chicago Press, Chicago & London: p. 189.
Šavrič, B., Jenny, B., Patterson, T., Petrovič D., Hurni, L. (accepted for publication). A polynomial
equation for the Natural Earth projection. Cartography and Geographic Information Science.
Wikipedia, the free encyclopedia - Newton’s method (2011). Wikimedia Foundation, Inc.
http://en.wikipedia.org/wiki/Newton-Raphson (accessed May 12, 2011).
8.2 Other references
Bildirici, I. Ö., Ipbüker, C., Yanalak, M. 2006. Function matching for Soviet-era table-based modified
polyconic projections. International Journal of Geographical Information Science 20, 7: 769–95.
52 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
Koler-Povh, T. 2011. Navodila za oblikovanje visokošolskih del na UL FGG in navajanje virov,
2. predelana izdaja. Ljubljana, Univerza v Ljubljani, Fakulteta za gradbeništvo in geodezijo: 59 p.
http://www3.fgg.uni-lj.si/uploads/media/UL_FGG_-
_Pr_10_Navodila_za_oblikovanje_visokosolskih_del_na_UL_FGG_04.pdf (accessed May 10, 2011).
Lampret, V., Premuš, M. 2007. Matematika II. Notes from the lectures and exercises. Ljubljana,
University of Ljubljana, Faculty of Civil and Geodetic Engineering.
Radovan, D. 2009. Kartografske projekcije. Notes from the lectures. Ljubljana, University of
Ljubljana, Faculty of Civil and Geodetic Engineering.
Stopar, B. 2007. Izravnalni račun II. Notes from the lectures. Ljubljana, University of Ljubljana,
Faculty of Civil and Geodetic Engineering.
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. 53
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
9 APPENDICES
APPENDIX A: Matlab code for the final determination of the polynimail coefficients
APPENDIX B: The code of Matlab function for the inverse transformation
APPENDIX C: Java Code for implementing the Natural Earth Projection
54 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
THIS PAGE IS INTENTIONALLY LEFT BLANK
TA STRAN JE NAMENOMA PRAZNA
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. A1
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
APPENDIX A:
MATLAB CODE FOR THE FINAL DETERMINATION OF THE POLYNOMAIL
COEFFICIENTS
This appendix provides the Matlab code, which was used for the final calculation of the polynomial
coefficients. The Matlab code takes into consideration all constraints to preserve the dimensions of the
graticule, and two additional measures to increase the smoothness of the rounded corners. Since the
used functional model is linear, no special loop is required for this determination.
% POLYNOMIAL APPROXIMATION FOR NATURAL EARTH PROJECTION
%
% Copyright [2011] [Bojan Savric]
%
% Licensed under the Apache License, Version 2.0 (the "License");
% you may not use this file except in compliance with the License.
% You may obtain a copy of the License at
%
% http://www.apache.org/licenses/LICENSE-2.0
%
% Unless required by applicable law or agreed to in writing, software
% distributed under the License is distributed on an "AS IS" BASIS,
% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
% See the License for the specific language governing permissions and
% limitations under the License.
%
% author: Bojan Savric, in collaboration with Tom Patterson, US National
% Park Service, and Bernhard Jenny, Institute of Cartography, ETH Zurich
%
%
% lengthP*s = A1 + A2*latitude^2 + A3*latitude^4 + A4*latitude^10 +
% + A5*latitude^12
% distanceP*s*k*Pi = B1*latitude + B2*latitude^3 + B3*latitude^7 +
% + B4*latitude^9 + B5*latitude^11
clear all;
clc
format long;
%STARTING DATA
s = 0.8707; %scale factor
k = 0.52; %height-to-width ratio
%on the whole defined area there are 37 points with known length and
%distance of parallels:
latitude = [-90, -85, -80, -75, -70, -65, -60, -55, -50, -45, -40, -35,...
-30, -25, -20, -15, -10, -5, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45,...
50, 55, 60, 65, 70, 75, 80, 85, 90]'*pi()/180; %unit is rad
lengthP = [0.55, 0.627, 0.6754, 0.716, 0.7525, 0.7874, 0.8196, 0.8492,...
0.8763, 0.9006, 0.9222, 0.9409, 0.957, 0.9703, 0.9811, 0.9894,...
0.9953, 0.9988, 1.0, 0.9988, 0.9953, 0.9894, 0.9811, 0.9703, 0.957,...
0.9409, 0.9222, 0.9006, 0.8763, 0.8492, 0.8196, 0.7874, 0.7525,...
0.716, 0.6754, 0.627, 0.55]'*s;
A2 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
distanceP = [-1.0, -0.9761, -0.9394, -0.8936, -0.8435, -0.7903, -0.7346,...
-0.6769, -0.6176, -0.5571, -0.4958, -0.434, -0.372, -0.31, -0.248,...
-0.186, -0.124, -0.062, 0.0, 0.062, 0.124, 0.186, 0.248, 0.31,...
0.372, 0.434, 0.4958, 0.5571, 0.6176, 0.6769, 0.7346, 0.7903,...
0.8435, 0.8936, 0.9394, 0.9761, 1.0]'*s*k*pi();
n = length(latitude);
%LEAST SQUARES METHOD
%matrix of coefficients for both polynomial approximations
% A_1 is for the length and A_2 is for the distance of parallels
for i=1:n
A_1(i,:)= [1, latitude(i)^2, latitude(i)^4, latitude(i)^10,...
latitude(i)^12];
A_2(i,:)= [latitude(i), latitude(i)^3, latitude(i)^7, latitude(i)^9,...
latitude(i)^11];
end
%LSM for the length of parallels
X0_1 = inv(A_1'*A_1)*A_1'*lengthP; %adjustment of indirect observations
d1 = length(X0_1);
%additional constraint - adding functionally dependence in LSM:
%(1) free coefficient A1 must be equal 1.0*s
C_1(1,d1)=0; C_1(1,1)=1; g1=lengthP(19);
M_1 = C_1*inv(A_1'*A_1)*C_1';
dX_1 = inv(A_1'*A_1)*C_1'*inv(M_1)*(g1-C_1*X0_1);
%unknowns with constraint included
X_1 = X0_1 + dX_1;
v_1 = A_1*X_1 - lengthP;
lengthP_1 = lengthP + v_1;
RMS(1) = sqrt(v_1'*v_1/(n - d1 + 1));
%LSM for the distance of parallels
X0_2 = inv(A_2'*A_2)*A_2'*distanceP; %adjustment of indirect observations
d2 = length(X0_2);
%additional constraints - adding functionally dependence in LSM:
%(2) the distance of parallels at 90° must be equal 1
%(3) the slope of this curve must be 7° at the latitude 90°
C_2(1,:)=[pi()/2, (pi()/2)^3, (pi()/2)^7, (pi()/2)^9, (pi()/2)^11];
C_2(2,:)=[1, 3*(pi()/2)^2, 7*(pi()/2)^6, 9*(pi()/2)^8, 11*(pi()/2)^10];
g2=[distanceP(n), tan(7*pi()/180)]';
M_2 = C_2*inv(A_2'*A_2)*C_2';
dX_2 = inv(A_2'*A_2)*C_2'*inv(M_2)*(g2-C_2*X0_2);
%unknowns with constraints included
X_2 = X0_2 + dX_2;
v_2 = A_2*X_2 - distanceP;
distanceP_2 = distanceP + v_2;
RMS(2) = sqrt(v_2'*v_2/(n - d2 + 2));
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. A3
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
%PLOTING THE RESULTS OF THE POLYNOMIAL APPROXIMATIONS
fprintf('\nPOLYNOMIAL APPROXIMATION FOR NATURAL EARTH PROJECTION');
fprintf('\n\nlengthP * s =\n');
fprintf('= A1 + A2*latitude^2 + A3*latitude^4 + A4*latitude^10 +
A5*latitude^12\n');
for j=1:d1
fprintf('A%d -> %9.6f\n', j, X_1(j));
end
fprintf('\ndistanceP * s*k*Pi =\n');
fprintf('= B1*latitude + B2*latitude^3 + B3*latitude^7 + B4*latitude^9 +
B5*latitude^11\n');
for j=1:d2
fprintf('B%d -> %9.6f\n', j, X_2(j));
end
A4 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
THIS PAGE IS INTENTIONALLY LEFT BLANK
TA STRAN JE NAMENOMA PRAZNA
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. B1
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
APPENDIX B:
THE CODE OF MATLAB FUNCTION FOR THE INVERSE TRANSFORMATION
This appendix provides the code of a Matlab function, which presents the inverse procedure for the
polynomial Natural Earth projection. The Newton-Raphson method is used, and it requires iterations
with while loop.
function [phi,lam]=NEP_xy2philam(X_p,Y_p,R)
%function [phi,lam]=NEP_xy2philam(X_p,Y_p,R) is the inverse transformation
%for the Natural Earth projection.
%
%Cartesian X/Y coordinates and radius R of the generating globe are
%required in meters. The output consist of spherical coordinates phi and
%lam in radians units.
%
% Copyright [2011] [Bojan Savric]
%
% Licensed under the Apache License, Version 2.0 (the "License");
% you may not use this file except in compliance with the License.
% You may obtain a copy of the License at
%
% http://www.apache.org/licenses/LICENSE-2.0
%
% Unless required by applicable law or agreed to in writing, software
% distributed under the License is distributed on an "AS IS" BASIS,
% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
% See the License for the specific language governing permissions and
% limitations under the License.
%
% author: Bojan Savric, in collaboration with Tom Patterson, US National
% Park Service, and Bernhard Jenny, Institute of Cartography, ETH Zurich
%DETERMINATION OF INITIAL GUESSES
Y_c = Y_p/R;
n = length(Y_c); %counting the numbers of points
%DETERMINATION OF LATITUDE
for i=1:n
x = Y_c(i);
%first approximation of solution
xn = x - ((1.007226.*x + 0.015085.*x.^3 - 0.044475.*x.^7 +...
0.028874.*x.^9 - 0.005916.*x.^11)-Y_c(i))/(1.007226 +...
0.045255.*x.^2 - 0.311325.*x.^6 + 0.259866.*x.^8 -
0.065076.*x.^10);
j=1;
%Newton-Raphson loop
while(abs(xn-x)>1e-13)
x = xn;
xn = x - ((1.007226.*x + 0.015085.*x.^3 - 0.044475.*x.^7 +...
0.028874.*x.^9 - 0.005916.*x.^11)-Y_c(i))/(1.007226 +...
0.045255.*x.^2 - 0.311325.*x.^6 + 0.259866.*x.^8 -...
B2 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
0.065076.*x.^10);
j=j+1;
if j==50 %exit in case of unending loop
fprintf('ERROR!!! example: %d\n', i);
break;
end
end
%final latitude value, and number of iteration steps
phi(i,1)=xn;
Js(i)=j;
end
%DETERMINATION OF LONGITUDE
one(1:n,1)= 0.8707;
L_phi = one - 0.131979.*phi.^2 - 0.013791.*phi.^4 + 0.003971.*phi.^10 -...
0.001529.*phi.^12;
lam = X_p./(R.*L_phi);
%PLOTING THE RESULTS OF INVERSE TRANSFORMATION
fprintf('\nTHE RESULTS OF INVERSE TRANSFORMATION FOR THE NATURAL EARTH
PROJECTION\n');
fprintf('\nThe numbers of iteration steps:\n');
for j=1:n
fprintf('point: %d -> %3d\n', j, Js(j));
end
fprintf('\nThe spherical coordinates of the Natural Earth projection:\n');
fprintf(' phi [rad] lam [rad]\n');
for j=1:n
fprintf('%3d: -> %10.6f %10.6f\n', j, phi(j), lam(j));
end
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. C1
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
APPENDIX C:
JAVA CODE FOR IMPLEMENTING THE NATURAL EARTH PROJECTION
This appendix provides the Java code implementing the final polynomial equation of the Natural Earth
projection in Flex Projector and in Java Map Projection Library (Jenny, B. 2011). The code contains
forward and inverse projections.
/*
Copyright 2011 Bojan Savric
Revised version of April 6, 2010.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
package com.jhlabs.map.proj;
import java.awt.geom.*;
/**
* The Natural Earth projection was designed by Tom Patterson, US National
Park
* Service, in 2007, using Flex Projector. The shape of the original
projection
* was defined at every 5 degrees and piece-wise cubic spline interpolation
was
* used to compute the complete graticule.
* The code here uses polynomial functions instead of cubic splines and
* is therefore much simpler to program. The polynomial approximation was
* developed by Bojan Savric, in collaboration with Tom Patterson and
Bernhard
* Jenny, Institute of Cartography, ETH Zurich. It slightly deviates from
* Patterson's original projection by adding additional curvature to
meridians
* where they meet the horizontal pole line. This improvement is by
intention
* and designed in collaboration with Tom Patterson.
*
* @author Bojan Savric
*/
public class NaturalEarthProjection extends PseudoCylindricalProjection {
private static final double A0 = 0.8707;
private static final double A1 = -0.131979;
private static final double A2 = -0.013791;
private static final double A3 = 0.003971;
C2 Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection.
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
private static final double A4 = -0.001529;
private static final double B0 = 1.007226;
private static final double B1 = 0.015085;
private static final double B2 = -0.044475;
private static final double B3 = 0.028874;
private static final double B4 = -0.005916;
private static final double C0 = B0;
private static final double C1 = 3 * B1;
private static final double C2 = 7 * B2;
private static final double C3 = 9 * B3;
private static final double C4 = 11 * B4;
private static final double EPS = 1e-11;
private static final double MAX_Y = 0.8707 * 0.52 * Math.PI;
@Override
public Point2D.Double project(double lplam, double lpphi,
Point2D.Double out) {
double phi2 = lpphi * lpphi;
double phi4 = phi2 * phi2;
out.x = lplam * (A0 + phi2 * (A1 + phi2 * (A2 + phi4 * phi2 * (A3 +
phi2 * A4))));
out.y = lpphi * (B0 + phi2 * (B1 + phi4 * (B2 + B3 * phi2 + B4 *
phi4)));
return out;
}
@Override
public boolean hasInverse() {
return true;
}
@Override
public Point2D.Double projectInverse(double x, double y, Point2D.Double
lp) {
// make sure y is inside valid range
if (y > MAX_Y) {
y = MAX_Y;
} else if (y < -MAX_Y) {
y = -MAX_Y;
}
// latitude
double yc = y;
double tol;
for (;;) { // Newton-Raphson
double y2 = yc * yc;
double y4 = y2 * y2;
double f = (yc * (B0 + y2 * (B1 + y4 * (B2 + B3 * y2 + B4 *
y4)))) - y;
double fder = C0 + y2 * (C1 + y4 * (C2 + C3 * y2 + C4 * y4));
yc -= tol = f / fder;
if (Math.abs(tol) < EPS) {
break;
}
}
lp.y = yc;
Šavrič, B. 2011. Derivation of a Polynomial Equation for the Natural Earth Projection. C3
Grad. Th. - University Studies. Ljubljana, UL FGG, Department of Geodetic Eng.
// longitude
double y2 = yc * yc;
double phi = A0 + y2 * (A1 + y2 * (A2 + y2 * y2 * y2 * (A3 + y2 *
A4)));
lp.x = x / phi;
return lp;
}
@Override
public String toString() {
return "Natural Earth";
}
}
