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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).

Copyright (c) Cartography and Geographic Information Society. All rights reserved.

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

Copyright (c) Cartography and Geographic Information Society. All rights reserved.

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).

Copyright (c) Cartography and Geographic Information Society. All rights reserved.

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

Copyright (c) Cartography and Geographic Information Society. All rights reserved.

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.

REFERENCES

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Copyright (c) Cartography and Geographic Information Society. All rights reserved.

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