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The Natural Earth Projection
The Natural Earth projection was
developed by Tom Patterson in 2007 out
of dissatisfaction with existing projections
for displaying physical data on small-scale world
maps (Jenny et al. 2008). Flex Projector, a
freeware application for the interactive design
and evaluation of map projections, was the
means for creating the Natural Earth projection.
The graphical user interface in Flex Projector
allows cartographers to adjust the length, shape,
and spacing of parallels and meridians of new
projections in a graphical design process (Jenny
and Patterson 2007).
A Polynomial Equation for the
Natural Earth Projection
Bojan Šavrič, Bernhard Jenny, Tom Patterson,
Dušan Petrovič, Lorenz Hurni
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 denes
the length and spacing of parallels in tabular form for every ve degrees of increasing latitude. It is
a pseudocylindrical projection, and is neither conformal nor equal-area. In the original denition,
piece-wise cubic spline interpolation is used to project intermediate values that do not align with the
ve-degree grid. This paper introduces alternative polynomial equations that closely approximate
the original projection. The polynomial equations are considerably simpler to compute and program,
and require fewer parameters, which should facilitate the implementation of the Natural Earth
projection in geospatial software. The polynomial expression also improves the smoothness of the
rounded corners where the meridians meet the horizontal pole lines, a distinguishing trait of the
Natural Earth projection that suggests to readers that the Earth is spherical in shape. Details on the
least squares adjustment for obtaining the polynomial formulas are provided, including constraints
for preserving the geometry of the graticule. This technique is applicable to similar projections that
are dened by tabular parameters. For inverting the polynomial projection the Newton-Raphson
root nding algorithm is suggested.
KEYWORDS: Projections, Natural Earth projection, Flex Projector
Cartography and Geographic Information Science, Vol. 38, No. 4, 2011, pp. 363-372
The Natural Earth projection is an amalgam of
the Kavraiskiy VII and Robinson projections,
with additional enhancements (Figure 1).
These two projections most closely fullled the
requirement for representing small-scale physical
data on world maps, but each had at least one
undesirable characteristic (Jenny et al. 2008).
The Kavraiskiy VII projection exaggerates the
size of high latitude 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 the map edges.
Creating the Natural Earth projection required
three major adjustments: Firstly, starting from
the Robinson projection, its vertical extension
was slightly increased to give it more height.
Secondly, using the Kavraiskiy VII as a template,
the parallels were slightly increased in length.
Bojan Šavrič and Dušan Petrovič, Faculty of Civil and Geo-
detic Engineering, University of Ljubljana, Slovenia, Email:
<bojansavric@gmail.com>, <dusan.petrovic@fgg.uni-lj.si>; Bern-
hard Jenny, College of Earth, Ocean and Atmospheric Sciences,
Oregon State University, Corvallis, Oregon, USA, Email: <jennyb@
geo.oregonstate.edu>; Tom Patterson, US National Park Service,
Harpers Ferry, West Virginia, USA, Email: <tom_patterson@nps.
gov>; Lorenz Hurni, Institute of Cartography and Geoinformation,
ETH Zürich, Switzerland, Email: <hurni@karto.baug.ethz.ch>.
DOI: http://dx.doi.org/10.1559/15230406384363
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Vol. 38, No. 4 364
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 result
of this procedure, the Natural Earth projection,
is a true pseudocylindrical projection, i.e., a
projection with regularly distributed meridians
and straight parallels (Snyder 1993:189). As
a compromise projection, the Natural Earth
projection is neither conformal nor equal area,
but its distortion characteristics are comparable
to other well known projections (Jenny et al.
2008). All three projections exaggerate the size
of high latitude areas (Figure 1). Appendix A
provides further details about the distortion
characteristics of the Natural Earth projection.
The shape of the graticule of any projection
designed with Flex Projector is dened 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. Equation 1 denes the original
Natural Earth projection, transforming spherical
coordinates into Cartesian X/Y coordinates, and
Table 1 provides the parameter values (Jenny et
al. 2008; 2010):
X = R ∙ s ∙ lᵩ ∙ λ lᵩ ∈ [0, 1], (Eq. 1)
Y = R ∙ s ∙ dᵩ ∙ k ∙ π dᵩ ∈ [-1, 1],
where:
X and Y are projected coordinates;
R is the radius of the generating globe;
s = 0.8707 is an internal scale factor;
lᵩ is the relative length of the parallel at latitude
φ, with φ ∈ [-π/2, π/2], lᵩ = 1 for the equator
and the slope of lᵩ is 63.883° at the poles;
dᵩ is the relative distance of the parallel at latitude
φ from the equator, with φ ∈ [-π/2, π/2] and
with dᵩ = ±1 for the pole lines, and dᵩ = 0 for
the equator;
λ is the longitude with λ ∈ [-π, π]; and
k = 0.52 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 dene lᵩ
and dᵩ in Equation 1 for every ve degrees. For
intermediate spherical coordinates that do not
align with the ve-degree grid, values for lᵩ and
dᵩ need to be interpolated. The Flex Projector
application uses a piece-wise cubic spline
interpolation, with each piece of the spline
curve covering ve degrees. While this type of
interpolation is rapid to evaluate, 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 efciency, the remainder of this paper
discusses a compact analytical expression that
approximates Equation 1 with two simple poly-
nomial expressions.
Analytical Expressions for the
Robinson Projection
Robinson and Patterson used an identical
approach for the design of their pseudocylindrical
projections. Both dened their projection by
Latitude
[degrees]
Relative length of
parallels
Relative distance of
parallels from equator
0 1 0
5 0.988 0.062
10 0.9953 0.124
15 0.9894 0.186
20 0.9811 0.248
25 0.9703 0.310
30 0.9570 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 0.7160 0.8936
80 0.6754 0.9394
85 0.6270 0.9761
90 0.5630 1
Table 1. Parameters for the Natural Earth projection: Rela-
tive lengths of parallels and relative distance from the
equator for every 5 degrees (after Jenny et al. 2008).
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365 Cartography and Geographic Information Science
adjusting the appearance of the projected ve-
degree graticule in an iterative process—Robinson
sketching the graticule with pen and paper, and
Patterson ne-tuning it in Flex Projector. In the
past, various authors have tackled the problem
of nding an analytical expression for the
Robinson projection. Since the two projections
are closely related, this section reviews existing
mathematical models of Robinson’s projection.
Polynomial approximation is recommended,
which is applied to the Natural Earth projection
in the next section.
Two general approaches exist for mathemati-
cally modeling graphically dened projections:
(1) interpolation and (2) approximation. The
Robinson projection has had both approaches
applied.
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 have used
interpolating methods for nding 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 difcult
implementation. For these reasons 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 for approximating the
Robinson projection (Equation 2). For the X
coordinates they use even powers up to the order
four, and for the Y coordinates odd powers up
to the order ve. Each expression contains three
coefcients, and the constants k, s and π of
Equation 1 are integrated with lᵩ and dᵩ. Their
solution contains only six parameters, and is fast
and simple to compute.
X = R ∙ λ ∙ (A0 + A2 ∙ φ2 + A4 ∙ φ4) (Eq. 2)
Y = R ∙ (A1 ∙ φ + A3 ∙ φ3 + A5 ∙ φ5)
where:
X and Y are projected coordinates;
φ and λ are the latitude and longitude;
R is the radius of the generating globe;
A0 = 0.8507;
A1 = 0.9642;
A2 = -0.1450;
A3 = -0.0013;
A4 = -0.0104; and
A5 = -0.0129.
A similar approach is proposed by Beineke
(1991; 1995). For lᵩ he suggests a polynomial
with even degrees up to the sixth order, and for
dᵩ he proposes an exponential approximation
with a real number exponent (Beineke 1991).
This approach uses a total of eight parameters
to approximate Robinson’s 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.
The approximating curves by Canters and
Decleir, as well as Beineke, use a smaller number
of parameters, and are considerably simpler
RobinsonKavraiskiy VII Natural Earth
Figure 1: The polynomial Natural Earth projection
compared to the Kavraiskiy VII and Robinson projections
(after Jenny et al. 2008).
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Vol. 38, No. 4 366
to program than the interpolating methods.
Polynomial equations are best in terms of
computation speed and code simplicity, but
higher-order terms might be necessary to
minimize deviations from the original curve.
Polynomial approximations, however, sometimes
suffer from undulations if the maximum degree is
too high, which must be avoided for a graticule to
appear smooth. Another potential drawback of
polynomial equations is the difculty of nding
inverse equations that transform from projected
X/Y coordinates to spherical coordinates. Indeed,
an analytical inverse does not generally exist for
higher-order polynomial equations. To solve for
spherical coordinates, numerical approximation
methods are necessary, such as the bisection or
the Newton-Raphson root nding algorithm.
A Polynomial Approximation for 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 coefcients 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
applications that project maps on the y 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
bounding meridians meet the horizontal pole
lines. It was found that the graphical tools and
the cubic spline interpolation in Flex Projector
do not provide sufcient control for dening
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. The new polynomial form of
the projection therefore deliberately deviates
from the original projection by adding curvature
to the corners. The changes to the smoothness
of the corners were entirely esthetic and done
to satisfy the authors’ sensibilities. They result
in a subjective improvement that cannot be
evaluated with objective criteria. Nor were they
applied for improving the projection’s distortion
characteristics.
The polynomial expression for the Natural
Earth projection is given in Equations 3 and 4.
The polynomials are of higher degrees than those
by Canters and Decleir (1989) for the Robinson
projection. Higher degrees are required for the
Natural Earth projection to smoothly model the
curved corners connecting the meridian lines to
the horizontal pole line.
X = R ∙ λ ∙ (A1 + A2 ∙ φ2 + A3 ∙ φ4 + A4 ∙ φ10 +
+ A5 ∙ φ12) (Eq. 3)
Y = R ∙ (B1 ∙ φ + B2 ∙ φ3 + B3 ∙ φ7 + B4 ∙ φ9 +
+ B5 ∙ φ11) (Eq. 4)
where:
X and Y are the projected coordinates;
φ and λ are the latitude and longitude in
radians;
R is the radius of the generating globe, and
A1 to A5 and B1 to B5 are coefcients given in
Table 2.
Equation 3 replaces both lᵩ and the factor s in
Equation 1 with a polynomial expression (with
A1 = s). The polynomials in Equation 4 include
the constant factors s, k, π, and dᵩ for reducing
the number of required multiplications and
accelerating calculations.
For developing Equations 3 and 4, the
following considerations were taken into
account: (1) The Natural Earth projection is
symmetrical about the x and y-axis; (2) it has
straight but not equally spaced parallels; and (3)
the parallels are equally divided by meridians.
Due to these characteristics, Equation 3 contains
only even powers of φ that are multiplied by λ,
and Equation 4 only consists of odd power terms
of φ (Canters, 2002, p. 133 ff.). For the purpose
of accelerating computations, the number of
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367 Cartography and Geographic Information Science
polynomial terms has been reduced. Equation 3
has no terms with degree 6 and 8, and Equation
4 has no term with degree 5.
When estimating the polynomial coefcients
with the method of least squares, two additional
constraints were added to bring the polynomial
graticule to the exact same size as the original
graticule. In Equation 3, the rst coefcient A1
was forced to equal the value of the internal
scale factor s (Equation 1). This is to ensure that
the length of the equator remains the same. In
Equation 4, the distance of the pole line from
the equator was forced to the original value by
introducing a second constraint.
Two additional measures were required to
increase the smoothness of the rounded corners
between the meridians and the pole lines. For
Equation 4, an additional constraint was added
to the method of least squares, xing the slope
of the polynomial to 7 degrees at the poles. The
second measure for improving the smoothness
of the corners involved slightly reducing the
length of the pole line before computing the
polynomial coefcients (Figure 2). The result is
a new polynomial Natural Earth projection that
deliberately deviates from the original projection
near the poles. Appendix B provides details on
the application of the method of least squares,
including the technique for integrating the
additional constraints, which should allow the
reader to apply this technique to other similar
projections.
Inverting the Polynomial
Natural Earth Projection
The inverse of a map projection transforms
Cartesian coordinates into spherical coordinates.
To determine the inverse of the polynomial
Natural Earth projection, Equations 3 and 4
must be inverted. The system dened by these
two polynomials has two known variables
(the Cartesian coordinates X and Y) and two
unknowns (the spherical coordinates φ and λ).
The system is solved by rst nding the latitude
φ in Equation 4, and then solving Equation 3 for
the unknown longitude λ.
An analytical expression of the inverse of
the polynomial Equation 4 does not exist, but
a large number of methods are available for
polynomial system solving (Elkadi and Mourrain
2005). The Newton-Raphson algorithm is a
numerical method for nding successively better
approximations to the roots or zeros of a real-
valued function, and is commonly used for the
numerical solving of nonlinear equations. The
Newton-Raphson root nding algorithm was
chosen for inverting the Natural Earth projection,
because it converges rapidly, is easy to compute,
and requires only one initial guess. Equation 5
shows the general form of the Newton-Raphson
algorithm.
xn+1 = xn – F ’(xn)-1 ∙ F(xn) F(xn) = 0 (Eq. 5)
where:
F(xn) and F’(xn) are a given function and its
derivative;
xn and xn+1 are the previous and the next
Coefcients for
Equation 3
Coefcients for
Equation 4
A10.870700 B11.007226
A2-0.131979 B20.015085
A3-0.013791 B3-0.044475
A40.003971 B40.028874
A5-0.001529 B5-0.005916
Table 2. Coefcients for the polynomial expression of the
Natural Earth projection.
A
B
C
A
B
Figure 2: The original (A) and polynomial (B) Natural Earth
projection are overlaid in (C). Arrows indicate changes in
smoothness at the end of the pole line, which is shortened
in (B).
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Vol. 38, No. 4 368
solution of the given function; and
n and (n+1) the steps of the iterative process.
The function F(xn) is formed by converting
Equation 4 to Equation 6:
F(xn) = B1 ∙ φ + B2 ∙ φ3 + B3 ∙ φ7 + B4 ∙ φ9 +
+ B5 ∙ φ11 – Y ∙ R-1 = 0 (Eq. 6)
The iterative approximation is repeated until a
sufciently accurate solution is reached. Con-
vergence to the solution is quadratic for Equa-
tion 6, since the derivative F’(xn) is positive for
all φ ∈ [-π/2, π/2], and F(xn) has therefore no
local minimum or maximum in the valid range
of φ. The closest local extremum is at φ = ±1.59,
which is outside the valid range of φ. The quo-
tient Y ∙ R-1 can be used as an initial guess for the
Newton-Raphson algorithm, as it is in the range
of the latitude φ, and does not have any local
extremum in this range (Equation 7).
Y ∙ R-1 ∈ [-s ∙ k ∙ π, s ∙ k ∙ π] ∈ [-π/2, π/2] (Eq. 7)
Applying the inverse projection of the polynomial
Natural Earth projection consists of the following
steps:
(1) The initial guess for the unknown latitude:
φ0 = Y ∙ R-1;
(2) With the Newton-Raphson approximation
method an improved latitude φ is calculated:
φn+1 = φn – F’(φn)-1 ∙ F(φn), where F(φn) is
the function from Equation 6, F’(φn) its
derivative, and n = 0, 1, 2, … , m. At step m
the iteration stops if | φm+1 – φm| < ε, where
ε is a sufciently small positive quantity,
typically close to the maximum precision of
oating point arithmetic;
(3) The nal latitude: φ = φm+1; and
(4) The nal longitude: λ = X ∙ R-1 ∙ (A1 +
+ A2 ∙ φ2 + A3 ∙ φ4 + A4 ∙ φ10 + A5 ∙ φ12)-1
The Newton-Raphson method is only
applied to compute the latitude φ in step (2);
the longitude λ can be computed in step 4 by
inverting Equation 3. The Newton-Raphson
method converges quickly with Equation 6. On
average, less than four iterations are needed
when transforming a regularly spaced graticule
with 15 degrees resolution covering the whole
sphere (with ε = 10-11).
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 φ and λ 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
is required (with an identical ε). However, the
Newton-Raphson method is faster, as it involves
fewer calculations, and is algorithmically simpler.
Conclusion
The Natural Earth projection expressed by the
polynomial Equations 3 and 4 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. This enhancement was developed in
collaboration with Tom Patterson, the author
of the original Natural Earth projection. The
polynomials are easy to code and fast to compute
as only seven multiplications are required for
each polynomial if factorized appropriately.
The Newton-Raphson method for inverting the
projection converges quickly, with only a few
iterations required. 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 identical to those
of the original projection. The areal distortion
and maximum angular distortion are similar
to those of other pseudocylindrical projections
(Appendix A). For these reasons, the authors
recommend using the polynomial equation of
the Natural Earth projection.
This article presents the development of
polynomial expressions for the Natural Earth
projection, which is one specic projection
designed with the graphical approach offered
by Flex Projector. Details on the least squares
adjustment with constraints for obtaining the
polynomial formulas are provided (Appendix B)
to allow others to apply this technique to similar
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369 Cartography and Geographic Information Science
projections dened by tabular parameters. It
remains to be explored how the polynomial
approximation method can be generalized for
any projection designed with Flex Projector.
ACKNOWLEDGEMENTS
This research was funded by the Erasmus student
exchange program, and was carried out at ETH
Zürich. The authors thank the anonymous
reviewers for their valuable comments.
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Vol. 38, No. 4 370
Appendix A:
Distortion Characteristics of the
Natural Earth Projection
As a compromise projection, the Natural Earth
projection is neither 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 projections, which
were used in the design procedure.
Figure 3 shows Tissot’s indicatrices for every
30 degrees. With increasing distance from the
equator the area of indicatrices increases, indi-
cating that the size of high latitude areas is exag-
gerated. Figure 3 omits indicatrices along pole
lines, since they are of innite size. The axes of
the indicatrices do not coincide with the direc-
tions of parallels and meridians, except at the
equator and the central meridian. Jenny et al.
(2008) present isocols of areal distortion for
the Natural Earth projection. Areal distortion
increases with latitude and does not change with
longitude (Table 3). All isocols of areal distor-
tion are therefore parallel to the equator. Areal
distortion is computed with σ = a
j · bj with aj
and bj the scale factors along the principal direc-
tions at position j on the sphere.
Angular distortion is moderate near the equator
and increases towards the edges of the graticule
(Table 4). Jenny et al. (2008) provide isocols of
maximum angular distortion. The values of max-
imum angular distortion ωj are constant along
the equator. Equation 8 computes ωj:
ωj = 2 arcsin aj - bj (Eq. 8)
aj + bj
where:
aj and bj are the scale factors along the principle
directions at position j on the sphere.
Figure 3. Tissot’s indicatrices for the polynomial Natural
Earth projection.
Latitude φ [̊] Area Scale
85 3.28
60 1.31
30 0.98
0 0.88
Table 3. Areal distortion increases with latitude.
φ
λ030 60 90 120 150 180
08.3 8.3 8.3 8.3 8.3 8.3 8.3
30 3.0 5.4 9.3 13.6 17.9 22.1 26.3
60 25.0 26.2 29.5 34.1 39.6 45.4 51.3
85 115.37 115.44 115.67 116.05 116.56 117.20 117.96
Table 4. Maximum angular distortion for every 30 degrees of
increasing latitude and longitude. All values are in degrees.
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371 Cartography and Geographic Information Science
Appendix B:
Least Squares Adjustment for Curve Fitting with Additional Constraints
This appendix presents the approximation method using least squares adjustment (LSA) with addi-
tional constraints. The presented approach is a modied LSA of indirect observations with functionally
dependent parameters (Mikhail and Ackerman 1976). It is hoped that the details provided here will
allow others to nd similar polynomial expressions for other projections.
The original Natural Earth projection is dened by 37 control points distributed over the complete
range of possible latitude values with φ ∈ [-π/2, π/2] and a control point every 5 degrees. For both
equations, polynomial expressions with ve terms were chosen. The functional model of the LSA is
given in Equation 9, which is derived from Equations 1, 3, and 4:
A1 + A2 ∙ φi2 + A3 ∙ φi4 + A4 ∙ φi10 + A5 ∙ φi12 = s ∙ lᵩ i (i, j = 1,… 37)
B1 ∙ φj + B2 ∙ φj3 + B3 ∙ φj7 + B4 ∙ φj9 + B5 ∙ φj11 = s ∙ k ∙ dᵩj ∙ π (Eq. 9)
The coefcients of the two polynomial expressions in Equation 9 are unknown parameters, and
the polynomial powers of latitude are known coefcients for the LSA. Equation 9 can be expressed in
matrix form (Equation 10), with n denoting the number of rows in the coefcients matrix A, and u the
number of parameters (n = 37, u = 5). The rst row of matrix A is [1, φ12, φ14, φ110, φ112] for the rst poly-
nomial, and [φ1, φ13, φ17, φ19, φ111] for the second polynomial. Vector x contains the parameters, i.e. the
unknown polynomial coefcients. In this particular case, no parameters are included in matrix A, which
results in a linear system that can be solved using the well known method of least squares in Equation
11. The vector v represents the minimized residuals after adjustment, for evaluating the standard devia-
tion of the model, and the differences between the original and the approximated graticule. The vector
l includes lᵩ and dᵩ, multiplied with the constant factors s, k and π as in Equation 9. Since this is a linear
model, no initial guess is required for the parameters.
An×u ∙ xu×1 = ln×1 (Eq. 10)
l + v = A∙x vT ∙ v min x = (AT∙A)-1 ∙ AT ∙ l v = A∙x – l (Eq. 11)
To initiate the constraints, an additional linear matrix equation is added to this model. For the Natu-
ral Earth projection three constraints were imposed: (1) the length of the parallel lᵩ at 0 degrees must
be 1; (2) the relative distance between the equator and the parallel dᵩ at 90 degrees must be 1; and (3)
the slope of the polynomial in Equation 4 is xed to 7 degrees at the pole line. All three constraints are
expressed with parameters, which is possible because the expressions in Equation 9 are linear.
For computing the polynomial coefcients of the relative length lᵩ 37 control points are used cover-
ing the whole range of possible latitude values between -π/2 and +π/2 with a distance of 5 degrees
between each pair of control points. The symmetrical arrangement of the control points around the
equator guarantees a continuously differentiable function.
The rst constraint (1) for the length lᵩ can be derived from Equation 9:
φ = 0 lᵩ = 1 A1 = s (Eq. 12)
The constraints (2) and (3) are applied to the relative distance dᵩ. The xed distance of the parallel at
90 degrees is expressed in a similar way as the constraint (1) in Equation 12. For the slope — constraint
(3)—a derivative of Equation 9 is used to express it. Both conditions are described in Equation 13.
φ = π/2 dᵩ = 1 B1 ∙ π/2 + B2 ∙ π3/8 + B3 ∙ π7/128 + B4 ∙ π9/512 + B5 ∙ π11/2048 = s ∙ k ∙ π
φ = π/2 B1 + 3 ∙ B2 ∙ π2/4 + 7 ∙ B3 ∙ π6/64 + 9 ∙ B4 ∙ π8/256 + 11 ∙ B5 ∙ π10/1024 = tan (7°) (Eq. 13)
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The additional constraints can be written in matrix form (Equation 14), where the vector x is the
same as in Equations 10 and 11. p is the number of additional constraints, which must be less than the
number of unknowns (p < u), as the model would otherwise become under-determined. The rst con-
straint for the lengths is represented in the matrix C and vector g by Equation 15. Equation 16 shows
the two matrixes for the distance constraints.
Cp×u ∙ xu×1 = gp×1 (Eq. 14)
C = [1 0 0 0 0], g = [ s ] (Eq. 15)
C = π/2 π3/8 π7/128 π9/512 π11/2048 (Eq. 16)
1 3∙π2/4 7∙π6/64 9∙π8/256 11∙π10/1024
g = s ∙ k ∙ π
tan (7°)
Equation 14 expresses the relationship between the functionally dependent parameters that must be
included in the LSA (Equation 11). This step is presented in Equation 17, where the two systems in
10 and 14 are solved together. The rst row in Equation 17 is a normal LSA of indirect observations.
The results are parameters not including the constraints. In the second row of Equation 17, the correc-
tions dx for the parameters are calculated. On the third row, the vector x is computed containing the
coefcients of the polynomial approximation. And nally, the vector of residuals v is computed. The
polynomial in x fullls all additional constraints expressed in Equation 14, and minimizes the deviations
from the curve dened by the 37 control points.
l + v = A ∙ x N = AT ∙ A x0 = N-1 ∙ AT ∙ l (Eq. 17)
C ∙ x = g M = C ∙ N-1 ∙ CT dx = N-1 ∙ CT ∙ M-1 ∙ (g - C ∙ x0)
x = x0 + dx v = A ∙ x – l
As Equation 9 is 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. The three constraints for the Natural Earth projection are linear, but non-linear constraints
could also be used. In this case, matrix C would contain partial derivatives of the constraints equations
with respect to all parameters in vector x, calculated from parameter values in vector x0. However, non-
linear constraints can only partially be fullled.
Vol. 38, No. 4 372
