Cartography, mathematical problems in

The problems arising in the construction of the mathematical foundations of (geographical and special) maps, that is to say, in the development of the theory of cartographic projections (also called map projections, cf. Cartographic projection), the study of their properties, transformations, surveying methods, and other matters. In this regard the surface of the Earth will be assumed to be either a sphere or an ellipsoid of revolution.

The basic object of study in mathematical cartography is the map projection: The representation, on the plane, of all or part of the terrestrial ellipsoid, given by the equations:

$$ \tag{1 } \left . \begin{array}{l} x = f _ {1} (u , v), \\ y = f _ {2} (u , v), \\ \end{array} \right \} $$

where $ u $ is the latitude and $ v $ the longitude of a point $ P \in \Delta $( $ \Delta $ being a simply-connected domain on the ellipsoid); $ x, y $ determine a point $ Q $ of the plane, called the image of $ P $; $ Q \in D $, where $ \Delta $ is mapped onto the domain $ D $. The functions $ f _ {1} $ and $ f _ {2} $ satisfy the following conditions: they are single-valued, twice continuously differentiable, and for the Jacobian of (1) one has $ h = \partial (x, y)/ \partial (u , v) \neq 0 $( in geodesy and cartography one always has $ h > 0 $, meaning that orientation is preserved). Most facts in mathematical cartography relate not only to mappings of an ellipsoid onto the plane, but also to mappings between arbitrary surfaces; therefore in the sequel a map projection (1) will mean a mapping of $ \Delta \subset S _ {1} $ onto $ D \subset S _ {2} $, where

$$ \tag{2 } S _ {1} : \ ds ^ {2} = \lambda _ {1} ^ {2} (u , v) [du ^ {2} + dv ^ {2} ]; $$

$$ S _ {2} : d \sigma ^ {2} = \lambda _ {2} ^ {2} (x, y) [dx ^ {2} + dy ^ {2} ] $$

are arbitrary regular surfaces, $ \Delta $ and $ D $ are simply-connected domains and $ ds, d \sigma $ are the corresponding line elements on the surfaces. The choice of the coordinate system is governed (apart from the simplicity of the description of the line element of the surface in terms of them and the advantage of their use in subsequent calculations) by the fact that transition to them from arbitrary curvilinear coordinates on the surface immediately provides a conformal mapping of the surface onto the plane (see [6], [3]). Even on an arbitrary surface, these coordinate systems are sometimes referred to as Cartesian coordinate systems (see [9]); in geodesy and cartography they are called isometric coordinate systems.

Knowledge of the mapping of $ S _ {1} $ onto $ S _ {2} $ given by (1) enables one to study the metric properties of the mapping (see [7], [2]), that is, to determine its characteristics: the scaling of the mapping given by $ \mu = d \sigma /ds $, where $ \mu = \mu (u , v; \alpha ) $, and, in particular, the scaling $ m $ along the lines on which $ v $ is constant, and the scaling $ n $ along the lines on which $ u $ is constant:

$$ m ^ {2} = \ \frac{\lambda _ {2} ^ {2} }{\lambda _ {1} ^ {2} } (x _ {u} ^ {2} + y _ {u} ^ {2} ), $$

$$ n ^ {2} = \frac{\lambda _ {2} ^ {2} }{\lambda _ {1} ^ {2} } (x _ {v} ^ {2} + y _ {v} ^ {2} ); $$

the rotations $ \psi $ and $ \chi $ of these lines:

$$ \mathop{\rm tan} \psi = \ \frac{y _ {u} }{x _ {u} } , $$

$$ \mathop{\rm tan} \chi = \frac{y _ {v} }{x _ {v} } ; $$

the angle $ \theta $ between the images of these lines:

$$ \theta = \chi - \psi ; $$

the angular deformation:

$$ \epsilon = \theta - { \frac \pi {2} } ; $$

the area change:

$$ p = \ \frac{\lambda _ {2} ^ {2} }{\lambda _ {1} ^ {2} } h; $$

the maximum angular distortion $ \omega $ given by

$$ \sin { \frac \omega {2} } = \ \frac{| a - b | }{a + b } , $$

where $ a $ and $ b $ are the relative scales for which

$$ a = \left ( { \frac \mu {M} } \right ) _ \max , $$

$$ b = \left ( { \frac \mu {M} } \right ) _ \min , $$

and $ M $ is the factor by which the ellipsoid has been reduced. The relative scales correspond to the principal directions which are orthogonal on both $ S _ {1} $ and $ S _ {2} $, and are related to $ m, n $ and $ \theta $ by Apollonius' theorems:

$$ a ^ {2} + b ^ {2} = \ m ^ {2} + n ^ {2} , $$

$$ ab = mn \sin \theta = p. $$

Taking $ A _ {0} $ to be the direction at an arbitrary point and $ \alpha _ {0} $ to be the image of this direction, one has

$$ \mathop{\rm tan} 2 \alpha _ {0} = \ \frac{2f }{e - g } $$

(where $ e = x _ {u} ^ {2} + y _ {u} ^ {2} $, $ f _ {v} = x _ {u} x _ {v} + y _ {u} y _ {v} $, $ g = x _ {v} ^ {2} + y _ {v} ^ {2} $), and

$$ \mathop{\rm tan} A _ {0} = \ { \frac{b}{a} } \ \mathop{\rm tan} \alpha _ {0} . $$

Associated with this is the Tissot indicatrix of the mapping, that is, the ellipse of distortion $ \{ a, b, A _ {0} , \psi \} $, regarded in the tangent plane to $ S _ {2} $ at the point $ Q $ as an ellipse similar to and similarly situated to the infinitely small ellipse (the principal part of the mapping) which is, up to $ o ( \rho ) $, the image of a circle of infinitely small radius $ \rho $, taken in the tangent plane to $ S _ {1} $ at the point $ P $. Of the above-mentioned characteristics, expressed in terms of the coefficients of the first fundamental forms of the mutually mapped surfaces (2) and the first-order partial derivatives of the mapping functions (1), only four are independent. The choice of a group of independent characteristics is not unique, and instead of $ \{ a, b, A _ {0} , \psi \} $ one often takes $ \{ m, n, \psi , \theta \} $.

In terms of the character of the distortions, the following map projections are distinguished: a) conformal or orthomorphic ( $ a = b $, $ \theta = \pi /2 $); b) equivalent or of equal area $ (ab = 1) $; c) equidistant or equal-spaced ( $ a = 1 $ or $ b = 1 $); d) geodesic, in cartography called orthodromic (an orthodrome is a geodesic on the sphere), under which geodesics on $ S _ {1} $ are taken into geodesics on $ S _ {2} $; and various other projections. The construction of projections having the combined properties of at least two of the above list a)–d) is possible only for a mapping of a surface onto a surface isometric to it, and in certain other trivial cases. Since conformality and equivalence are not compatible conditions for the purposes of cartography, equidistant projections which, with regard to the character of the distortions they give, occupy an intermediate position between a) and b), have a special significance. The selection of a separate class of map projections is usually dictated by inherent properties of the projections, e.g. of the types a)–c), that is, by the representation of the equations for their characteristics, which with regard to the formulas of distortion theory are easily transformed into first-order partial differential equations. The set of solutions of a concrete system of two such equations describes a specific class of projections. The projections are ascribed the type corresponding to the type of these equations (elliptic, hyperbolic, etc.).

The formulas of distortion theory given above enable one to make the appropriate choice of the projection needed for some concrete purpose: having established by some means the equations (1) (for example, by giving on the plane the images of the coordinate lines of the ellipsoid and, possibly, by making use of some additional conditions such as conformality or equivalence, etc.), one can then study the mapping based on these formulas and, by varying its parameters, select the most suitable projection for drawing the map of the given domain $ \Delta \subset S _ {1} $. In addition to this solution to the direct problem, use is also made of other methods (geometric, graphical-analytic, etc.).

The inverse problem of cartography is to find projections on the basis of distortions assigned a priori to them (see [11]). Here the question of the existence of the required projections is of prime importance. The answer to this question is provided by the fundamental system of equations in the theory of mapping of surfaces (see [4]):

$$ \tag{3 } \left . \begin{array}{l} m _ {v} ^ {*} - n _ {u} ^ {*} \cos \theta + \psi _ {u} n ^ {*} \sin \theta + \theta _ {u} n ^ {*} \sin \theta = 0, \\ \psi _ {v} m ^ {*} - n _ {u} ^ {*} \sin \theta - \psi _ {u} n ^ {*} \ \cos \theta - \theta _ {u} n ^ {*} \cos \theta = 0, \\ \end{array} \right \} $$

in which

$$ m ^ {*} = \ \frac{\lambda _ {1} (u , v) }{\lambda _ {2} (x, y) } m, $$

$$ n ^ {*} = \frac{\lambda _ {1} (u , v) }{\lambda _ {2} (x, y) } n. $$

For a mapping of a surface onto a plane, $ \lambda _ {2} (x, y) \equiv 1 $, and for a mapping of one plane domain onto another, $ \lambda _ {1} (u , v) \equiv 1 $ as well. The system (3) is not well-defined: four independent characteristics of the mapping are related by two equations for it; the various methods of making the system well-defined and of interpreting it allow for a large variety of applications. The system (3) is quasi-linear with respect to any pair of characteristics. The following two theorems hold.

Theorem 1) The characteristics $ m, n, \psi , \theta $ of a mapping of a domain $ \Delta \subset S _ {1} $ onto a domain $ D \subset S _ {2} $ given by a set of single-valued twice continuously-differentiable functions (1) with Jacobian of constant sign in $ \Delta $, satisfy (3) at all points of this domain.

Theorem 2) Let $ \Delta $ be a simply-connected domain on a given regular surface and let $ \phi _ {i} = \phi _ {i} (u , v) $ be four given functions defined and continuous on $ \Delta $ along with their first-order partial derivatives and taking their values in some simply-connected domain $ \Pi $ belonging to the $ 4 $- dimensional parallelepiped

$$ \{ 0 < \phi _ {1} < k _ {1} ; \ 0 < \phi _ {2} < k _ {2} ; \ k _ {i} = \textrm{ const } ,\ i = 1, 2; $$

$$ {}- \pi < \phi _ {3} \leq \pi ; 0 < \phi _ {4} < \pi \} . $$

If these functions are taken as the characteristics of some mapping of $ \Delta \subset S _ {1} $ onto the plane

$$ m = \phi _ {1} ,\ \ n = \phi _ {2} ,\ \ \psi = \phi _ {3} ,\ \ \theta = \phi _ {4} , $$

and if they satisfy (3) throughout $ \Delta $, then a mapping can be reconstructed from them which takes $ \Delta \subset S _ {1} $ homeomorphically onto some domain $ D $ in the $ xy $- plane, which is twice continuously differentiable, has Jacobian $ h > 0 $ in $ \Delta $, and takes an arbitrary point $ (u _ {0} , v _ {0} ) \in \Delta $ into a given point $ (x _ {0} , y _ {0} ) $ of the $ xy $- plane.

Theorem 2 gives, for a given distribution of the characteristics, conditions for the existence of a mapping of a simply-connected domain $ \Delta $ on $ S _ {1} $ onto some domain $ D $ on $ S _ {2} $ such that some interior point $ (u _ {0} , v _ {0} ) $ is mapped to $ (x _ {0} , y _ {0} ) $ in the plane; here the boundary of $ D $ is unknown. Lack of knowledge concerning the mapped domain is characteristic for mathematical problems in cartography: the mapped domain is determined either after setting up the mapping functions (1), or it has to be found on the basis of supplementary conditions, e.g., conditions for minimizing the distortion under projection.

For mappings of the sphere onto the plane, the system (3) turns into the so-called Euler–Urmaev system (see [5], [11]), while for mappings of plane domains, when the system (3) is overdetermined by a system of equations in the characteristics, it gives as a consequence a derived system of quasi-conformal mappings (see [12]). The reduction of mappings of surfaces to quasi-conformal mappings of plane domains (with bounded distortion) is completely natural, and the former, that is, the mappings (1) of the given domain $ \Delta \subset S _ {1} $ onto the given domain $ D \subset S _ {2} $, can be interpreted as "triple" projections: the domain $ \Delta \subset S _ {1} $ is conformally mapped onto a domain $ \widetilde \Delta $ of the $ uv $- plane, the domain $ D \subset S _ {2} $ is conformally mapped onto a domain $ \widetilde{D} $ of the $ xy $- plane and the domain $ \widetilde \Delta $ of the $ uv $- plane is quasi-conformally mapped onto the domain $ \widetilde{D} $ of the $ xy $- plane. The quasi-conformal mapping between the plane domains $ \widetilde \Delta $, $ \widetilde{D} $ associated with the mapping from $ \Delta \subset S _ {1} $ onto $ D \subset S _ {2} $, has the following characteristics: $ V = m ^ {*} $, $ \alpha = \psi $, $ W = n ^ {*} \sin \theta $, where the angle $ \theta $ has the same meaning as before. The relationship between these mappings makes it possible (see [4]) to apply the theory of quasi-conformal mappings of plane domains to mathematical problems in cartography. In this connection, the system of equations of any class of cartographic projections or, more generally, any non-linear system of differential equations of the form

$$ \tag{4 } F _ {i} (u , v, x _ {u} , x _ {v} , y _ {u} , y _ {v} ) = 0,\ \ i = 1, 2, $$

can, after writing it in the characteristics

$$ \tag{5 } J _ {i} (u , v, m ^ {*} , n ^ {*} , \psi , \theta ) = 0,\ \ i = 1, 2, $$

be reduced to the quasi-linear system which follows from (3) after overdetermining it by the system (5), which is regarded as a derived system of the original system (4). However, the relation between the types of these two systems, the original and the derived, remains an open question.

The apparatus of quasi-conformal mappings of plane domains (with two pairs of characteristics) can also be drawn upon as the theoretical basis of instrumental transformations of map projections (with the aid of mechanical, optical, electrical, and other devices); more specifically, of a transformation of one simply-connected plane domain onto another, each of which is the result of drawing a map of the same domain of the terrestrial ellipsoid, but under different projections.

One of the basic problems in cartography (in the area of the creation of small-scale maps and partially medium-scale maps) is the problem of optimal map projections, that is, those projections of a given domain $ \Delta $ on a surface onto the plane under which the distortion is (in some prescribed sense) minimal (see [2]). The most popularized criteria for deciding the merits of map projections are the following:

the Airy criterion:

$$ \Phi _ {A} = \ \int\limits _ \Delta \epsilon _ {E} ^ {2} \ d \Delta ,\ \ \epsilon _ {A} ^ {2} = \ { \frac{1}{2} } [(a - 1) ^ {2} + (b - 1) ^ {2} ]; $$

the Jordan criterion:

$$ \Phi _ {J} = \ \int\limits _ \Delta \epsilon _ {J} ^ {2} \ d \Delta ,\ \ \epsilon _ {J} ^ {2} = \ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } ( \mu - 1) ^ {2} \ d \alpha ; $$

the Airy–Kavraiskii criterion:

$$ \Phi _ {A - K } = \ \int\limits _ \Delta \epsilon _ {E - K } ^ {2} \ d \Delta ,\ \ \epsilon _ {A - K } ^ {2} = \ { \frac{1}{2} } ( \mathop{\rm ln} ^ {2} a + \mathop{\rm ln} ^ {2} b); $$

the Jordan–Kavraiskii criterion:

$$ \Phi _ {J - K } = \ \int\limits _ \Delta \epsilon _ {J - K } ^ {2} \ d \Delta ,\ \ \epsilon _ {J - K } ^ {2} = \ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} ^ {2} \mu d \alpha $$

(these are all of variational type), and

the Chebyshev criterion (which is a criterion of minimax type), according to which the quality of a projection is given either by

$$ \delta = \ \frac{\sup a \mid _ \Delta }{\inf b \mid _ \Delta } , $$

by its logarithm or by the maximum modulus of this logarithm. Depending on which type of criterion is used in the construction of a map projection, the optimal map projections of variational and minimax type will be distinct. Thus, the problems of constructing optimal map projections consist in the requirement that for a given domain $ \Delta $ on a surface $ S $ one seeks (subject to minimizing one of the above functionals $ \Phi $ or the quantity $ \delta $) a projection of $ \Delta $ onto the plane either from the entire set (1) of projections, such projections being called ideal, or from some subset of them (for example, from the class of projections described by the solutions of the corresponding system (4)), such projections being called best projections within this subset. For each such problem it is necessary to determine whether the problem is well-posed.

For a spherical segment (a spherical region bounded by a small circle), an ideal projection with respect to the Chebyshev criterion (see [2], [13]) is the equidistant projection of Postel; the best conformal one is the stereographic (see [8]); the best equivalent one is the equal-area projection of Lambert (see [2]). On the basis of variational criteria, G.B. Airy (see [14]) calculated the best azimuthal projection for a spherical segment. For a spherical trapezium — a spherical region bounded by two meridians and two parallels — and regions approximating a spherical trapezium (in some sense), the conical projections are used in map drawing.

Of the projections intended for mapping spherical trapezia, the best projections of minimax type are known (Markov projections, see [10]), and also the best conformal and equivalent ones (of the same type) (see [2]). A general method has been developed for the construction of best conical (conformal, equivalent, equidistant) projections according to the Airy criterion, for simply-connected domains of arbitrary shape (see [15], [2]). However, for the case of a spherical trapezium mapped onto the plane by a conformal projection, the smallest fluctuation of the logarithmic scale occurs for projections calculated (see [9], [11]) in accordance with the Chebyshev–Grave theorem (see [8], [9]): In order that for a conformal projection ( $ a = b $, $ \epsilon = 0 $) of a simply-connected domain $ \Delta $ on a surface $ S $ with total curvature of constant sign onto the plane, the logarithmic scale has minimum deviation from zero, it is necessary and sufficient that the scale is constant on the contour of the mapped territory. Conformal projections satisfying the conditions of this theorem are called Chebyshev projections. They possess a number of useful properties (for them $ ( \sup \mu | _ \Delta )/( \inf \mu | _ \Delta ) = \min $, $ \Phi _ {J - K } = \min $( the mean curvature of the mapped geodesics of the surface $ S $ is minimal), and are therefore of value in practice. P.L. Chebyshev solved the problem of finding for $ \Delta \subset S $ a conformal projection with minimal distortion of the area, that is a conformal projection closest to an equivalent projection.

The converse problem is also important: To find among all equivalent projections of a domain $ \Delta \subset S $ onto the plane, a projection with minimal distortion of shape. One of the approaches to this problem is the following. In view of the fact that these two sets of projections have different cardinalities (the set of conformal projections is described by a system of two partial differential equations $ m = n $, $ \epsilon = 0 $, while for the set of equivalent projections there is only one, $ mn \cos \epsilon = 1 $), one first of all selects from the class of equivalent projections a certain class of projections which are close to conformal ones, and then within this class one looks for a projection with minimal distortion of shape. One such class of projections, of classical heritage, is known, namely the class of Euler projections: $ ab = 1 $, $ \epsilon = 0 $( see [5]); nowadays other classes of equivalent projections close to conformal projections have been proposed, for example the class: $ ab = 1 $, $ (m - n) + k \epsilon = 0 $, where $ k = k (u , v) $ is a parameter of the class (see [4]).

The study of these and other new classes of map projections and the search for optimal projections of minimax type within them, has led to the statement of conditional minimax problems (see [16]). The essence of such problems is that for a given system of partial differential equations whose solutions have to be found in the given domain $ \Delta $, it is required to determine extra conditions (boundary, initial, etc.), and also in certain cases to determine the shape of those curves to which these conditions have to relate, such that on integrating the system subject to these conditions, one of the solutions in the domain $ \Delta $ will have minimal deviation from zero. Only special cases are known for the study of such problems, based on the method of a priori estimates of solutions of differential equations. Results obtained for systems of elliptic type are an extension of the Chebyshev–Grave theorem to two new classes of projections: the class described by the system of equations in the characteristics $ n = km $, $ \epsilon = 0 $, $ k = \textrm{ const } > 0 $( see [17]), and the class $ m = n ^ {c} $, $ \epsilon = 0 $, $ c = \textrm{ const } > 0 $( see [19]). For hyperbolic systems describing Euler projections, initial conditions on the meridian (see [16]) and on the parallel (see [18]) have been established by the method of characteristics, under which one of the solutions of the system (the logarithmic scale) has minimal deviation from zero in the domain of influence of the initial data. The cartographic meaning of this analogue of Chebyshev's theorem is the following: the "carrier" of the Cauchy data must be the lines of conformality, that is, the relative scale on it must be equal to one. This result is valid also, under given initial conditions on the meridian, for the wider class of projections: $ m = n ^ {c} $, $ \epsilon = 0 $, $ c = \textrm{ const } $( see [19]). The case of Euler projections has been investigated for a given spherical trapezium; here, in considering the mixed problem (invoking the energy integral), initial (for the South side of the trapezium) and boundary (on the arcs of the meridians bounding it) conditions are established which ensure that the best Euler projection of variational type is obtained for this domain: on the above three sides of the domain the relative scale must be equal to one (see [20]). The general method of determining optimal projections of variational type for an arbitrary domain $ \Delta \subset S $ reduces to the solution of variational problems with free boundaries (see [4]).

References

[1]	Itogi Nauk. i Tekhn. Kartografiya , 7 (1976) pp. 45–57
[2]	V.V. Kavraiskii, "Collected works" , 2 : 1–3 , Moscow (1958–1960) (In Russian)
[3]	V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 1–2 , Moscow-Leningrad (1947–1948) (In Russian)
[4]	G.A. Meshcheryakov, "Theoretical foundations of mathematical cartography" , Moscow (1968) (In Russian)
[5]	L. Euler, "Selected articles on cartography" , Moscow (1959) (In Russian; translated from German)
[6]	C.F. Gauss, "Selected work on geodesy" , 2 , Moscow (1958) (In Russian)
[7]	M.A. Tissot, "Mapping of one surface onto another and the composition of geographic maps" , Moscow (1899) (In Russian; translated from French)
[8]	P.L. Chebyshev, "Sur la construction des cartes géographiques" , Oeuvres de P.L. Chebyshev , 1 , Chelsea, reprint (1962) pp. 233–236; 239–247
[9]	D.A. Grave, "On the fundamental problems of the mathematical theory of constructing geographical maps" , Handbook math. libr. (1896) (In Russian)
[10]	A.A. Markov, Izv. Akad. Nauk SSSR Ser. 5 , 2 : 3 (1895) pp. 177–187
[11]	N.A. Urmaev, "Methods for finding new cartographic projections" , Moscow (1947) (In Russian)
[12]	M.A. Lavrent'ev, "Variational methods for boundary value problems for systems of elliptic equations" , Noordhoff (1963) (Translated from Russian)
[13]	J. Milnor, "A problem in cartography" Amer. Math. Monthly , 76 (1969) pp. 1101–1112
[14]	G.B. Airy, Philos. Mag. Ser. 4 , 22 (1861) pp. 409–421
[15]	N.Ya. Tsinger, Izv. Akad. Nauk SSSR Ser. 6 , 10 : 17 (1916) pp. 1693–1704
[16]	G.A. Meshcheryakov, "New extremal problems" Soviet Math Dokl. , 2 (1961) pp. 158–161 Dokl. Akad. Nauk SSSR , 136 : 5 (1961) pp. 1026–1029
[17]	M.A. Topchilov, Izv. Vuzov. Geod. i Aerofotos'emka , 4 (1970) pp. 91–96
[18]	Ya.I. Tuchin, Trudy Novosibirsk. Inst. Inzhenerov Geod., Aerofotos'emki i Kartografii , 30 (1973) pp. 65–67
[19]	Yu.M. Yuzefovich, Sb. Nauchn. Trudov Belorussk. Sel'khoz. Akad. , 86 (1972) pp. 245–253
[20]	Ya.I. Tuchin, Trudy Novosibirsk. Inst. Inzhenerov Geod. Aerofotos'emki i Kartografii , 34 (1973) pp. 55–64

Comments

In Western literature the phrase map projection is preferred to cartographic projection. It always establishes a correspondence between points of the ellipsoid and an $ xy $- plane (see also (1)). Instead of $ u , v $ one more often uses $ \phi , \lambda $ for latitude and longitude. The coordinates $ x, y $ are called map coordinates.

A geodesic, or orthodrome, is a line giving a path of shortest distance between two points on a surface (see also Geodesic line). The phrase "geodesic projection" is seldom used in Western literature. A well-known map projection not mentioned in the main article above is the gnomonic projection. It is a perspective projection of the globe (ball) from the centre onto a plane tangent at some point on the surface.

For references see Cartographic projection.

References

[a1]	A.H. Robinson, "Elements of cartography" , Wiley (1960)