[ Table of contents ]

5        Abstract coordinate systems

5.1  Introduction

An abstract coordinate system is a means of identifying positions in position-space by coordinate n-tuples. An abstract coordinate system is completely defined in terms of the mathematical structure of position-space. In this International Standard the term “coordinate system”, if not otherwise qualified, is defined to mean “abstract coordinate system.” Each coordinate system has a coordinate system type (see 5.4). Other coordinate system related concepts defined in this clause include coordinate-component surfaces and coordinate-component curves (see 5.5), linearity and other properties (see 5.6), and localization (see 5.7). Map projections and augmented map projections are defined and treated as special cases of the general abstract coordinate system concept (see 5.8). Standardized abstract coordinate systems are specified in 5.9.

In Clause 6 a temporal coordinate system is defined as a means of identifying events in the time continuum by coordinate 1-tuples using an abstract coordinate system of coordinate system type 1D. In Clause 8 a spatial coordinate system is defined as an abstract coordinate system suitably combined with a normal embedding (see Clause 7) as a means of identifying points in object-space by coordinate n-tuples.

5.2  Preliminaries

This International Standard takes a functional approach to the construction of coordinate systems. Annex A provides a concise summary of mathematical concepts and specifies the notational conventions used in this International Standard. In particular, Annex A defines the terms interior, one-to-one, smooth, smooth surface, smooth curve, orientation-preserving, and connected. The concept of Rn as a vector space, the point-set topology of Rn, and the theory of real-valued functions on Rn are all assumed. Algebraic and analytic geometry, including the concepts of point, line, and plane, are also assumed. Together with such common concepts, a newly introduced concept replete will be used. A set D is replete if all points in D belong to the closure of the interior of D (see Annex A). A replete set is a generalization of an open set that allows the inclusion of boundary points. Boundary points are important in the definitions of certain coordinate systems.

5.3  Abstract CS

An abstract Coordinate System (CS) is a means of identifying a set of positions in an abstract Euclidean space that shall be comprised of:

a)       a CS domain,

b)       a generating function, and

c)       a CS range,

where:

a)       The CS domain shall be a connected replete domain in the Euclidean space of n-tuples (1  n  m), called the coordinate-space.

b)       The generating function shall be a one-to-one, smooth, orientation-preserving function from the CS domain onto the CS range.

c)       The CS range shall be a set of positions in a Euclidean space of dimension m (n  m  3), called the position-space. When n = 2 and m = 3, the CS range shall be a subset of a smooth surface5. When n = 1 and m = 2 or 3, the CS range shall be a subset of an implicitly specified smooth curve6.

An element of the CS domain shall be called a coordinate7. The kth-component of a coordinate n-tuple (1  k  n) may be called the kth coordinate-component. Coordinate-component8 is the collective term for any kth coordinate-component.

An element of the CS range shall be called a position. The coordinate of a position p shall be the unique coordinate whose generating function value is p.

The generating function may be parameterized. The generating function parameters (if any) shall be called the CS parameters.

The inverse of the generating function shall be called the inverse generating function. The inverse generating function is one-to-one and is smooth and orientation-preserving in the interior of its domain. A CS may equivalently be defined by specifying the inverse generating function when the CS domain is an open set.

NOTE 1       The generating function of a CS is often specified by an algebraic and/or trigonometric description of a geometric relationship (see 5.3 Example). There are also CSs that do not have geometric derivations. The Mercator map projection (see Table 5.18) is specified to satisfy a functional requirement of conformality (see 5.8.3.2) rather than by a geometric construction.

EXAMPLE            Polar CS: Considering the polar geometry depicted in Figure 5.1, define a generating function F as:

formula or figure

where:

formula or figure

The CS domain of F in coordinate-space is formula or figure.

The CS range of F in position-space is formula or figure.

This generating function is illustrated in Figure 5.2. The grey boxes with lighter grey edges in this figure represent the fact that the range in position-space extends indefinitely, and that the domain in coordinate-space extends indefinitely along the r-axis. The dotted grey edges indicate an open boundary. This CS range, CS domain, and generating function define an abstract CS representing polar coordinates as defined in mathematics [EDM, “Coordinates”]. The normative definition of the polar CS may be found in Table 5.33.

formula or figure

Figure 5.1Polar CS geometry


formula or figure

Figure 5.2The polar CS generating function

NOTE 2        In the special case where 1) the CS domain and CS range are both Rn and 2) the function is the identity function, this approach to defining coordinate systems reduces to the usual definition of the Euclidean coordinate system on Rn where each point is identified by an n-tuple of real numbers [EDM] (see Table 5.8, Table 5.29 and Table 5.35).

NOTE 3        The CS generating function has an inverse because it is one-to-one, but the inverse may be discontinuous at points in the image of CS domain boundary points. This is the case for the positive x-axis in the example above.

5.4  CS types

The coordinate-space and position-space dimensions characterize an abstract CS by CS type as defined in Table 5.1.

Table 5.1 — CS types

CS type

Dimension of coordinate-space

Dimension of position-space

3D

3

3

surface

2

3

curve9

1

3

2D

2

2

plane curve9

1

2

1D

1

1

 

A CS of CS type 3D may be called a 3D CS, a CS of CS type surface may be called a surface CS, and a CS of CS type 2D may be called a 2D CS.

5.5  Coordinate surfaces, induced surface CSs, and coordinate curves

5.5.1        Introduction

The generating function of a 3D CS is a function of the three coordinate-components of a coordinate 3-tuple. If one of the coordinate-components is held fixed (to a constant value), then the generating function thus restricted to two variables may be viewed as a surface CS generating function (with a surface CS range). If two of the three coordinate-components are held fixed, the generating function restricted to one variable may be viewed as a curve CS generating function (with curve CS range). These observations motivate the definitions of coordinate-component surfaces and curves. The coordinate-component surface and coordinate-component curve concepts are required to specify induced CS relationships, for the definition of special coordinate curves parallel and meridian, and the definition of CS handedness (see also 10.5).

5.5.2        Coordinate-component surfaces and induced surface CSs

If F is the generating function of a 3D CS, and u = (u0v0w0) is in the interior of the CS domain D, then three surface CS generating functions at u are defined by:

              formula or figure,

              formula or figure, and

              formula or figure.

The CS domain for S1 is the connected component of formula or figurewhich contains (v0w0).
The CS domain for
S2 is the connected component of formula or figurewhich contains (u0w0).
The CS domain for
S3 is the connected component of formula or figurewhich contains (u0v0).

Each of these surface CSs shall be called, respectively, the 1st, 2nd, and 3rd surface CS induced by F at u.

The CS ranges of these surface CSs are, respectively, the 1st, 2nd, and 3rd coordinate-component surface at u.

EXAMPLE 1         Coordinate-component surface: The geodetic 3D CS with generating function formula or figureis specified in Table 5.14 with CS parameters a and b. The 3rd coordinate-component surface at formula or figureis the surface of the oblate ellipsoid with major semi-axis a and minor semi-axis b.

EXAMPLE 2         Induced surface CS: The surface geodetic CS is specified in Table 5.24. Its CS domain, CS range and generating function are identical to the 3rd surface CS induced by the geodetic 3D generating function at formula or figure. If h is replaced by 0 in the formulae for the generating and inverse generating functions of the geodetic 3D CS, they reduce to the surface geodetic formulae.

5.5.3        Coordinate-component curves

Coordinate-component curves are defined for CSs of CS type 3D, CS type surface, and CS type 2D.

The CS type 3D case:

If F is the generating function of a CS of CS type 3D, D is the CS domain, and u = (u0v0w0) is in the interior of D, then the 1st, 2nd, and 3rd coordinate-component curves at u are parametrically specified, respectively, by the following smooth functions:

              formula or figure,

              formula or figure, and

              formula or figure.

The domain for C1 is the connected component of formula or figurewhich contains u0.
The domain for
C2 is the connected component of formula or figurewhich contains v0.
The domain for
C3 is the connected component of formula or figurewhich contains w0.

NOTE          The intersection of two coordinate surfaces at u is (the locus of) a coordinate-component curve: formula or figure

The CS type surface and CS type 2D cases:

If F is the generating function of a CS of CS type surface or CS type 2D, D is the CS domain, and u = (u0v0) is in the interior of D, then the 1st and 2nd coordinate-component curve at u is parametrically specified, respectively, by the following smooth functions:

              formula or figure, and

              formula or figure.

The domain for C1 is the connected component of formula or figurewhich contains u0.
The domain for
C2 is the connected component of formula or figurewhich contains v0.

EXAMPLE            If formula or figureis in the interior of the CS domain of the polar CS generating function F of the 5.3 Example, then the first coordinate-component curve isformula or figure, and the 2nd coordinate-component curve isformula or figure.

If F is the generating function for the geodetic 3D CS or the surface geodetic CS, and formula or figurein the 3D case or formula or figurein the surface case, then (see Figure 5.3):

a)       the 1st coordinate-component curve at u shall be called the parallel at u, and

b)       the 2nd coordinate-component curve at u shall be called the meridian10 at u.

The meridian at formula or figureshall be called the prime meridian11.

The parallel at formula or figureshall be called the equator.

formula or figure

Figure 5.3Geodetic 3D CS geometry, and coordinate-component surface and curves

5.6  CS properties

5.6.1        Linearity

A CS with generating function F is a linear CS if F is an affine function. The CS domain of a linear coordinate system is all of the coordinate-spaceformula or figure.

A curvilinear CS is a non-linear CS.

EXAMPLE            The polar CS of 5.3 EXAMPLE is a curvilinear CS of CS type 2D.

5.6.2        Orthogonality

A CS of CS type 3D, CS type surface, or CS type 2D is orthogonal if the angle between any two coordinate-component curves at u is a right angle when u is any coordinate in the interior of the CS domain of the generating function.

EXAMPLE            The polar CS of 5.3 EXAMPLE is a orthogonal CS of CS type 2D.

5.6.3        Linear CS properties: Cartesian, and orthonormal

In a linear CS, the kth coordinate-component curve is a line. The kth coordinate-component curve at the origin 0 of a linear CS is the kth-axis.

In a linear CS, if the angles between coordinate-component curves at the origin 0 are (pair-wise) right angles, then that is the case at all points. In particular, a linear CS is orthogonal12 if the axes are orthogonal.

In some publications a Cartesian CS is defined the same way as an orthogonal linear CS13. This International Standard, however, defines this concept differently. A linear CS with generating function F is a Cartesian CS if formula or figure (i.e., the axis unit points are all one unit distant from the origin F(0)).

An orthonormal CS is a linear CS that is both orthogonal and Cartesian.

A CS of CS type 3D with generating function F is orientation-preserving if the Jacobian determinant of F is positive.

EXAMPLE            The Lococentric Euclidean 3D CS specified in Table 5.9 is an orientation-preserving orthonormal CS.

5.6.4        CS right-handedness and coordinate-component ordering

Given a CS of CS type 3D and a coordinate formula or figure in the interior of the CS domain, the coordinate-component curves at p determine an ordered set of three tangent vectors:

formula or figure

An orthogonal CS of CS type 3D is a right-handed CS if for some coordinate formula or figure in the interior of the CS domain, the ordered set of tangent vectors t1, t2, and t3 form a right-handed coordinate system as defined in IS0 31-11. The right-handed CS property is determined, in part, by the order of the coordinate-components in the coordinate 3-tuple. The order of the coordinate-components in the specification of an orthogonal CS of CS type 3D shall be restricted to an ordering that insures a right-handed CS. This restriction is required for uniform treatment of directions in an SRF (see 10.5).

The coordinate-component ordering in the specification of a surface CS that is induced on a coordinate-component surface of a 3D CS, shall use the coordinate-component order of the inducing 3D CS.

EXAMPLE 1         The geodetic 3D CS (Table 5.14) coordinate-component ordering formula or figure insures that the CS is right-handed. A similar ordering for the planetodetic 3D CS (Table 5.15) is not right-handed because the tangent to planetodetic longitude points opposite to the direction of the tangent to geodetic longitude. Instead, the coordinate-component ordering formula or figure is specified to satisfy the right-handed CS requirement.

EXAMPLE 2         The surface planetodetic geodetic CS (Table 5.25)) coordinate-component ordering formula or figure is determined by the coordinate-component ordering formula or figure of the planetodetic 3D CS (Table 5.15).

5.7  CS localization

In some applications of a CS in the context of a spatial reference frame, it is necessary to consider a modified version of the CS that has been translated to a local origin and/or rotated (see "Lococentric" spatial reference frame variants in Clause 8). To treat these modifications in a uniform manner, the generating function of a CS that has been translated to a local origin and/or rotated is related to the generating function of the original CS by means of a localization operator. This uniform method, defined below, of specifying the variant CS by composing the original CS generating function with a localization operator shall be called CS localization.

Three parameterized operators, called localization operators, that operate on or between position-spaces are defined in Table 5.2. The inverses of these operators are defined in Table 5.3.

Table 5.2 — Localization operators

Localization
operator

Domain

Range

Localization parameters

Operator definition

formula or figure

formula or figure

formula or figure

q, r, s, in formula or figure
r and s are orthonormal

formula or figure

formula or figure

formula or figure

formula or figure

q, r, s, in formula or figure
r and s are orthonormal

formula or figure

formula or figure

formula or figure

formula or figure

q, r, s, in formula or figure
r and s are orthonormal

formula or figure

 

Table 5.3 — Localization inverse operators

Localization operator

Inverse operator definition

formula or figure

formula or figure

formula or figure

formula or figure

formula or figure

formula or figure

 

There are several forms of CS localization depending on CS type and localization operator. A 3D or surface CS with generating function F is localized by composing F with the formula or figurelocalization operator. The localized CS is of the same CS type (CS type 3D or CS type surface, respectively). Its generating function is formula or figure and has the same CS domain as F.

There are two localization operators for a 2D CS. One uses localization parameters in formula or figure and produces a surface CS. The other uses localization parameters in formula or figure and produces a 2D CS.

a.       A 2D CS with generating function FC is localized by composing F with the formula or figurelocalization operator. The localized CS is a surface CS. Its generating function is formula or figure and has the same CS domain as F.

b.       A 2D CS with generating function F is localized by composing F with the formula or figurelocalization operator. The localized CS is a surface CS. Its generating function is formula or figure and has the same CS domain as F.

The localization operator parameter q shall be called the lococentre. A localized CS may be called a lococentric CS.

NOTE          CS localization preserves the following CS properties: linear/curvilinear, orthogonal, Cartesian, and orthonormal.

The relationship between a CS type and its localized version(s) is summarized in Table 5.4.

Table 5.4 — Localized CS type relationships

CS type

Localization operator

Lococentric CS type

3D

formula or figure

3D

Surface

formula or figure

Surface

2D

formula or figure

formula or figure

2D

 

5.8  Map projection coordinate systems

5.8.1        Map projections

Map projections are 2D models of a 3D curved surface. In this International Standard, map projections are limited to the surface of an oblate ellipsoid. A map projection (MP) is comprised of

a)       an MP domain in the surface of an oblate ellipsoid,

b)       a generating projection, and

c)       an MP range in 2D coordinate-space,

where:

a)       the MP domain is a connected subset of the surface of the oblate ellipsoid,

b)       the MP range is a connected replete set, and

c)       the generating projection is one-to-one from the MP domain in the oblate ellipsoid onto its MP range and its inverse function is smooth and orientation-preserving in the MP range interior.

NOTE 1       This definition may be generalized to any ellipsoid including tri-axial ellipsoids, but this International Standard only addresses map projections for oblate ellipsoids.

NOTE 2       The domain of a map projection is always a proper subset of the oblate ellipsoid surface. In particular, the domain of the Mercator map projection (see Table 5.18) omits the pole points.

The generating projection P is specified in terms of surface geodetic CS coordinates (see Table 5.24). The component functions formula or figure and formula or figure of the generating projection P shall be called the mapping equations:

              formula or figure

where:

              formula or figure

The MP range coordinate-components u and v shall be called easting and northing, respectively. The positive direction of the u-axis (the easting axis) shall be called map-east. The positive direction of the v-axis (the northing axis) shall be called map-north.

The inverse mapping equations are the component functions formula or figure and formula or figure of the inverse generating projectionformula or figure:

              formula or figure

5.8.2        Map projection as a surface CS

If the inverse generating projection of a map projection Q is composed with the surface geodetic CS generating functionformula or figure, the resulting function formula or figure is the generating function of a surface CS (see Figure 5.4). The CS domain is the MP range. In this International Standard, a map projection CS shall be a surface CS for which the generating function is implicitly specified in terms of the mapping equations of a map projection.

In some cases, the surface geodetic coordinates with coordinate-component formula or figure are not in the MP domain of P nor are they in the range of Q. However, if the composite function formula or figureis continuous at the pole pointsformula or figure, then formula or figureand formula or figureshall be extended by continuity to include the pole points in the CS range.

NOTE          The CS generating function formula or figureis not to be confused with the generating projection P.

 

formula or figure

Figure 5.4The generating function of a map projection

5.8.3        Map projection geometry

5.8.3.1            Introduction

In general, the Euclidean geometry that a surface CS 2D coordinate-space inherits from formula or figure has no direct significance with respect to the geometry of position-space. In particular, the Euclidean distance between a pair of surface geodetic coordinates has no obvious meaning in position-space. In contrast, map projections are specifically designed so that coordinate-space geometry will model one or more geometric aspects of the corresponding oblate ellipsoid surface in position-space.

The map projection CSs specified in this International Standard are designed so that one or more geometric aspects of the MP domain in the oblate ellipsoid surface are approximated or modelled by the corresponding aspect in coordinate-space. The length of the line segment between two map coordinates is related to the length of the corresponding surface curve. Similarly, one or more of directions, areas, the angles between two intersecting curves, and shapes may be related approximately or exactly to the corresponding geometric aspect on the oblate ellipsoid surface.

The extent to which these aspects are or are not closely related is an indication of distortion. Some map projection CSs are designed to eliminate distortion for one geometric aspect (such as angles or area). Others are designed to reduce distortion for several geometric aspects. In general, distortion tends to increase with the size of the oblate ellipsoid MP domain relative to the total oblate ellipsoid surface area. Conversely, distortion errors may be reduced by restricting the size of the MP domain. Map projections specified in this International Standard in the context of a spatial reference frame may have areas of definition beyond which the projection should not be used for some application domains due to unacceptable distortion14.

5.8.3.2            Conformal map projections

A conformal map projection preserves angles. For such map projections, when two surface curves on an oblate ellipsoid meet at the angle a, the image of those curves in the map coordinate-space meet at the same angle a [THOM].

In addition, [THOM] contains a derivation based on the theory of complex variables to obtain conditions that specify when a projection is conformal. The map projections specified in Table 5.18 through Table 5.22 are conformal. The equidistant cylindrical MP specified in Table 5.23 is not conformal.

NOTE          The conformal property is local. A conformal map projection preserves angles at a point, but does not necessarily preserve shape or area. In particular, a large projected triangle may appear distorted under a conformal map projection.

5.8.3.3            Point distortion

One indicator of map projection length distortion is the ratio of lengths between an infinitesimal line segment in coordinate-space and the corresponding curve in position-space. Given a point in the interior of the MP range with surface geodetic coordinate (l,f) the directional point distortion15 at (l,f) with respect to a smooth surface curve passing through the point is the ratio of the differential distance in coordinate-space to the differential arc length at (l,f) along the curve as determined by the mapping equations.

The latitudinal point distortion at (l,f), denoted j(l,f), is the directional point distortion with respect to the meridian at (l,f). It is computed in the direction of the meridian at the point as:

formula or figure

whereformula or figureis the radius of curvature in the meridian as specified in Table 5.6.

The longitudinal point distortion at (l,f), denoted k(l,f), is the directional point distortion with respect to the parallel at (l,f). It is computed in the direction of the parallel at the point as:

formula or figure

whereformula or figureis the radius of curvature in the prime vertical as specified in Table 5.6.

If a map projection is conformal, then the directional point distortion is independent of the direction of the curve at the point. In particular, j(l,f) = k(l,f) for conformal map projections.

It is common practice in cartography to convert map projection coordinate-space to a display coordinate-space by means of a scaling factor. The scaling factor s  shall be termed a map scale [HTDP] and a point in the display space shall be termed a display coordinate16. The relationship of a display coordinate (ud, vd) to a map coordinate (u, v) is:

formula or figure

Map scale is commonly expressed as a ratio 1:n.

EXAMPLE            A map scale printed on a map sheet as 1:50 000 corresponds to s = 1/50 000.

For a conformal map projection, the infinitesimal ratio of display distance to arc length along a parallel is the point scale at (l,f) and is denoted by kscaled. The relationship between point scale and point distortion is:

kscaled(l,f) = s k(l,f).

 

5.8.3.4            Geodetic azimuth and map azimuth

The geodetic azimuth17 from a non-polar point p1 on the surface of an ellipsoid to a second point p2 on the surface is the angle measured clockwise from the meridian curve segment connecting p1 to the North pole to the geodesic containing p1 and p2 (see Figure 5.5). The range of azimuth values formula or figure shall be formula or figure. The definition and range constraints apply to points in both hemispheres.

In a map projection CS, the map azimuth from a coordinate c1 to a coordinate c2 is defined as the angle from the v-axis (map-north) clockwise to the line segment connecting c1 to c2. In general, the map azimuth for a pair of coordinates will differ in value from the geodetic azimuth of the corresponding points on the oblate ellipsoid.

formula or figure

Figure 5.5Geodetic azimuths a12 from p1 to p2 and a34 from p3 to p4



5.8.3.5            Convergence of the meridian

Given a point formula or figure in the interior of the MP domain of a map projection, the meridian through that point is projected to a curve in coordinate-space that passes through the corresponding coordinate. The angle g at the coordinate in the clockwise direction from the curve to the v-axis (map-north) direction shall be called the convergence of the meridian (COM) (see Figure 5.6).

The relationship formula or figureis used to derive the formulae for COM from the mapping equations of each of the map projections18. The COM angle is adjusted to the rangeformula or figure.

NOTE          If the map projection is conformal, then an equivalent relationship is given by: formula or figure.

A typical geometry illustrating the COM at a point p is shown for the transverse Mercator map projection in Figure 5.6.

formula or figure

Figure 5.6Convergence of the meridian

EXAMPLE            If p2 is directly map-north of p1 (it has a larger v coordinate-component), then the map azimuth is zero, but the geodetic azimuth may not be zero. The geodetic azimuth is approximately the sum of the map azimuth and the COM if the points are sufficiently close together.

5.8.4        Relationship to projection functions

5.8.4.1            Projection functions

Projection functions are defined in A.9. In some cases, the generating projection of a map projection CS is derived from a projection function. The derivation involves two steps. The first step is to restrict the domain of the projection function to a specified region of a given oblate ellipsoid so that the restricted function is one-to-one. The range of a projection function is a surface in 3D position-space. The second step is to associate that surface to 2D coordinate-space without introducing additional distortions.

In the case of planar projection functions, including the orthographic, perspective, and stereographic projection functions, the range is in a plane that can be identified with 2D coordinate-space by selecting an origin and unit axis points.

In the case of the cylindrical and conic projection functions, the range surface is a cylinder or a cone, respectively. These surfaces are developable surfaces and, except for a line of discontinuity, are homeomorphic to a subset of 2D coordinate-space with a homeomorphism that has a Jacobian determinant equal to one. Conceptually, these surfaces can be unwrapped to a flat plane without stretching the surface.

The polar stereographic MP (Table 5.22) is derived from the stereographic projection function, and in the spherical case, it is a conformal map projection. The same derivation may be applied to an oblate ellipsoid. However, the resulting map projection will not have the conformal property. For this reason, the generalization of the polar stereographic map projection mapping equations from the spherical case to the non-spherical oblate ellipsoid case is not derived from the polar stereographic projection function. Instead it is derived analytically to preserve the conformal property. Similarly, the Mercator map projection (Table 5.18) is designed to have the conformal property and is not derived from the cylindrical projection function even in the case of a sphere.

EXAMPLE            Polar stereographic: Given a sphere with a polar point p, the tangent plane to the sphere at p and the opposite polar point v specify a stereographic planar projection function F (see A.9.2.3). The restriction of F to a subsurface of the sphere that excludes v, is the generating projection for the sphere case of a polar stereographic map projection. In Figure 5.7 the position s on the sphere is projected to point t on a plane.

formula or figure

Figure 5.7Polar stereographic map projection

5.8.4.2            Map projection classification

The use of projection functions to derive map projections with desirable properties is limited, but does motivate some classifications of map projections. These classifications include tangent and secant map projections as well as conic and cylindrical map projections [SNYD, p. 5].

5.8.4.3            Cylindrical map projections

A map projection is classified as cylindrical if:

a)       all meridians of the oblate ellipsoid project to parallel straight lines that are equally-spaced with respect to the longitude of the meridians, and

b)       all parallels of the oblate ellipsoid project to parallel straight lines that are perpendicular to the meridian images.

As a consequence, formula or figure for a cylindrical map projection.

EXAMPLE            The Mercator map projection (Table 5.18) and the equidistant cylindrical map projection (Table 5.23) are both classified as cylindrical map projections.

A cylindrical map projection is tangent if the longitudinal point distortion is equal to one along the equator. It is secant if the longitudinal point distortion is equal to one along two parallels equally-spaced from the equator in latitude. In that case, the parallel with positive latitude shall be called the standard parallel. Tangent and secant cylindrical map projections are illustrated in Figure 5.8.

formula or figure

Figure 5.8Tangent and secant cylindrical map projections

5.8.4.4            Conic map projections

A map projection is classified as conic if:

a)       all meridians of the oblate ellipsoid project to radial straight lines that are equally-spaced in radial angle with respect to the longitude of the meridians, and

b)       all parallels of the oblate ellipsoid project to concentric arcs that are perpendicular to the meridian images.

As a consequence, g depends on l only for conic map projections because the projections of meridians are straight line segments with an angle depending only on the longitude.

EXAMPLE             Lambert conformal conic (see Table 5.21) is classified as a conic projection.

A conic map projection is tangent if along one parallel the longitudinal point distortion is equal to one. It is secant if the longitudinal point distortion is equal to one along two parallels that are not symmetric about the equator. In that case, the two parallels shall be called the standard parallels (see Figure 5.9).

formula or figure

Figure 5.9Tangent and secant conical map projections

5.8.5        Map projection CS common parameters

To avoid negative coordinate-component values or to reduce the magnitude of the values in a region of interest in the coordinate-space of a map projection, two mapping equation parameters shall be provided to control the position of the coordinate-space originformula or figure. One, denoted as uF, shall be called the false easting and shall offset easting values. The second, denoted as vF, shall be called the false northing and shall offset northing values.

A map projection CS specification may specify additional mapping equation parameters. The mapping equations parameters, including false easting and false northing shall be also be called the CS parameters.

The CS parameters longitude of origin, denoted by formula or figure and latitude of origin, denoted by formula or figure are historically associated with some map projection CSs. Typically, the position with map coordinate formula or figurehas geodetic longitude equal toformula or figure and/or geodetic latitude equal toformula or figure.

If formula or figureis present as a CS parameter in a map projection CS specification, it is used as the longitudinal centring function parameter (see formula or figure in Table 5.6.)

The CS parameter central scale, denoted by formula or figurewhen present, is intended to control the tangent/secant characteristics of the map projection CS and is therefore close to, but does not generally exceed, 1,0. In the case of a sphere, formula or figurecorresponds to a tangent projection, and formula or figure corresponds to a secant projection. Typically, some point in the MP domain with geodetic longitude equal to formula or figure and/or geodetic latitude equal to formula or figure will have a longitudinal point distortion equal toformula or figure.

For some MP cases, if formula or figure, there will exist a parallel with longitudinal point distortion equal to 1 at each point on the parallel. The latitude of such a parallel is termed a secant latitude, or a latitude of true scale.

NOTE          Central scale should not be confused with Map scale. Map scales (see 5.8.3.3) are typically much smaller in magnitude and are applied directly to the coordinate-space. In particular, if a transverse Mercator map projection with central scale value k = 0,9996 is to be scaled 50 000:1 on a map sheet display, then the mapping equations formula or figure are evaluated with k0 = 0,9996 and the display coordinates formula or figureare plotted on the map sheet with s = (1/50 000).

5.8.6        Augmented map projections

5.8.6.1            Augmentation with ellipsoidal height

A 3D CS can be specified from a map projection CS. The canonical embedding of a point (u, v) in R2 to the point (u, v, 0) in the uv-plane of R3 allows map points in 2D coordinate-space to be augmented with a third coordinate axis, the w-axis of R3. To be considered as a 3D CS, an augmented 3-tuple (uvw) of coordinates in the augmented map projection coordinate-space shall be associated to a unique position in position-space. The association is to ellipsoidal heightformula or figure. Given an augmented coordinate-tuple (uvw) for which (uv) belongs to the coordinate range of the underlying generating projection, the associated position is given in 3D geodetic coordinates (lf, h) where (lf,) is projected to (uv) by the map projection mapping equations. The third coordinate-space coordinate w is the vertical coordinate and the 3D geodetic coordinate constraints on negative values of h impose corresponding constraints on allowed values for w. In some application domains, other vertical coordinate measures are used (see Clause 9). Augmentation is restricted to ellipsoidal height in this International Standard.

5.8.6.2            Distortion in augmented map projections

In addition to map projection distortion (see 5.8.3.1), augmentation causes additional distortion. Consider the two straight-line segments between the pairs of coordinate-space points {(u1, v1, 0), (u2, v2, 0)} and {(u1, v1, w), (u2, v2, w)} with w > 0 (see Figure 5.10). In augmented map projection geometry, the two line segments have the same length. The corresponding curve in position-space of the first line segment is a surface curve of the oblate ellipsoid (or sphere). The corresponding second curve is outside of the oblate ellipsoid (or sphere) and has longer arc length than the first, and the length difference increases with w.

In general, vertical angles are not preserved and the angular error will vary with the point distortion value. These and other distortions have profound implications for dynamic equations that are beyond the scope of this International Standard.

formula or figure

Figure 5.10 Vertical distortion

5.9  CS specifications

5.9.1        Specification table elements and common functions and parameters

The CSs specified in this International Standard are presented in Table 5.8 through Table 5.35. Each CS specification specifies the values of all elements presented in Table 5.5.

Table 5.5 — Coordinate system specification elements

Element

Definition

Description

A description of the CS including a common name, if any.

CS label

The label of the CS (see 13.2.2).

CS code

The code of the CS (see 13.2.3).

Function type

Either “generating function” or “map projection”.

CS descriptor

One of: 3D linear, 3D curvilinear, surface linear, surface curvilinear, map projection, 2D linear, 2D curvilinear, 1D linear, 1D curvilinear, or surface (map projection) and 3D (augmented map projection).

Properties

Either “none” or a list of one or more properties of the CS chosen from the following: orthogonal, not orthogonal, orthonormal, not orthonormal, conformal, or not conformal. Conformal and not conformal only apply to map projections.

CS parameters and constraints

The CS parameters (if any) along with any constraints on how those parameters interrelate, otherwise “none”.

Coordinate-components

Coordinate-component symbols and common names in a specified order.

Domain of the generating function or mapping equations

The domain of the CS generating function or the mapping equations.

Generating function or mapping equations

The CS generating function or mapping equations.

Domain of the inverse of the generating function or mapping equations

The domain of the inverse of the CS generating function or the domain of the inverse of mapping equations.

Inverse of the generating function or mapping equations

The inverse of the CS generating function or the inverse of mapping equations.

COM

For map projection CSs, the equation for g in radians. Otherwise "n/a".

Point distortion

For map projection CSs, the equation for k if conformal or
the equation for
k and j equations if non-conformal. Otherwise "n/a".

Figures

Zero or more figure(s) that explain and illustrate the CS.

Notes

Optional, non-normative information concerning the CS, otherwise "none".

References

The references (see 13.2.5).

 

A specific ordering of coordinate-components for a coordinate n-tuple in a CS specification is required in this International Standard for clarity of presentation, to avoid ambiguity in the specification of the API, and, in the case of an orthonormal CS of CS type 3D, to insure the right-handed CS property (see 5.6.4). Alternatively, a coordinate-component value may be identified by its coordinate-component name and/or symbol in place of its identification by order in a coordinate-tuple.

Several specified CS generating functions and mapping equations and/or their inverses use some common intermediate functions or parameters associated with oblate ellipsoids. For clarity and concise presentation, these functions and parameters are defined in Table 5.6.

Table 5.6Common parameters and functions of an oblate ellipsoid

Function or parameter

Symbol and defining expression

major semi-axis

formula or figure

minor semi-axis

formula or figure

flattening

formula or figure

(first) eccentricity

formula or figure
alternative equivalent expression:

formula or figure

second eccentricity

formula or figure

radius of curvature in the prime vertical

formula or figure

radius of curvature in the meridian

formula or figure
alternative equivalent expression:
formula or figure

meridional distance to equator

formula or figure

longitudinal centring about formula or figure 

formula or figure

 

NOTE 1       Replacing formula or figure with formula or figure gives the inverse of the longitudinal centring function. That is:

                           formula or figure

NOTE 2       The function arctan2, used in many CS specification tables, is defined in A.8.2.

Table 5.7 presents a directory of the CS specifications in this International Standard. Each listed CS is specified in a separate table that is indicated by the hyperlink in the corresponding cell in the “Table number” column. Additional CSs may be added by registration (see Clause 13).

Table 5.7CS specification directory

Function
type

CS type

Label and Description

Table
number

Generating function

3D

EUCLIDEAN_3D
Euclidean 3D

Table 5.8

LOCOCENTRIC_EUCLIDEAN_3D
Lococentric Euclidean 3D

Table 5.9

SPHERICAL
Spherical

Table 5.10

LOCOCENTRIC_SPHERICAL
Lococentric spherical

Table 5.11

AZIMUTHAL_SPHERICAL
Azimuthal spherical

Table 5.12

LOCOCENTRIC_AZIMUTHAL_SPHERICAL
Lococentric azimuthal spherical

Table 5.13

GEODETIC
Geodetic 3D

Table 5.14

PLANETODETIC
Planetodetic 3D

Table 5.15

CYLINDRICAL
Cylindrical

Table 5.16

LOCOCENTRIC_CYLINDRICAL
Lococentric cylindrical

Table 5.17

Map projection

Surface and augmented 3D

MERCATOR
Mercator

Table 5.18

OBLIQUE_MERCATOR_SPHERICAL
Oblique Mercator spherical

Table 5.19

TRANSVERSE_MERCATOR
Transverse Mercator

Table 5.20

LAMBERT_CONFORMAL_CONIC
Lambert conformal conic

Table 5.21

POLAR_STEREOGRAPHIC
Polar stereographic

Table 5.22

EQUIDISTANT_CYLINDRICAL
Equidistant cylindrical

Table 5.23

Generating function

Surface

SURFACE_GEODETIC
Surface geodetic

Table 5.24

SURFACE_PLANETODETIC
Surface planetodetic

Table 5.25

LOCOCENTRIC_SURFACE_EUCLIDEAN
Lococentric surface Euclidean

Table 5.26

LOCOCENTRIC_SURFACE_AZIMUTHAL
Lococentric surface azimuthal

Table 5.27

LOCOCENTRIC_SURFACE_POLAR
Lococentric surface polar

Table 5.28

2D

EUCLIDEAN_2D
Euclidean 2D

Table 5.29

LOCOCENTRIC_EUCLIDEAN_2D
Lococentric Euclidean 2D

Table 5.30

AZIMUTHAL
Azimuthal

Table 5.31

LOCOCENTRIC_AZIMUTHAL
Lococentric azimuthal

Table 5.32

POLAR
Polar

Table 5.33

LOCOCENTRIC_POLAR
Lococentric polar

Table 5.34

1D

EUCLIDEAN_1D
Euclidean 1D

Table 5.35

 

5.9.2        Euclidean 3D CS specification

Table 5.8 — Euclidean 3D CS

Element

Specification

Description

Euclidean 3D.

Label

EUCLIDEAN_3D

Code

1

Function type

Generating function.

CS descriptor

3D linear.

Properties

Orthonormal.

CS parameters and constraints

none

Coordinate-components

u, v, w

Domain of the generating function or mapping equations

R3

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

R3

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

Coordinate-space 3-tuples are identified with position-space 3-tuples.

References

[EDM]

 

5.9.3        Lococentric Euclidean 3D CS specification

Table 5.9Lococentric Euclidean 3D CS

Element

Specification

Description

Localized Euclidean 3D

Label

LOCOCENTRIC_EUCLIDEAN_3D

Code

2

Function type

Generating function.

CS descriptor

3D linear.

Properties

Orthonormal.

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R3, and
      
r, s: axis directions in R3.

Constraints:
      
r and s are orthonormal vectors.

Coordinate-components

u, v, w

Domain of the generating function or mapping equations

R3

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

R3

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

1)       Euclidean 3D CS (see Table 5.8) is a special case withformula or figure.

2)       The generating function is the composition of the generating function for Euclidean 3D CS (see Table 5.8) with the 3D localization operator (see 5.7).

References

[EDM]

 

5.9.4        Spherical CS specification

Table 5.10 — Spherical CS

Element

Specification

Description

Spherical.

Label

SPHERICAL

Code

3

Function type

Generating function.

CS descriptor

3D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

none

Coordinate-components

l : longitude in radians,
q : spherical latitude in radians, and
r : radius.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

The spherical CS is not intrinsically associated with any specific sphere.

References

[HCP]

 

5.9.5        Lococentric spherical CS specification

Table 5.11 — Lococentric spherical CS

Element

Specification

Description

Localized spherical

Label

LOCOCENTRIC_SPHERICAL

Code

4

Function type

Generating function

CS descriptor

3D curvilinear

Properties

Orthogonal

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R3, and
      
r, s: axis directions in R3.

Constraints:
      
r and s are orthonormal.

Coordinate-components

l : longitude in radians,
q : spherical latitude in radians, and
r : radius.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

The generating function is the composition of the generating function for spherical CS (see Table 5.10) with the 3D localization operator (see 5.7).

References

[EDM]

 

5.9.6        Azimuthal spherical CS specification

Table 5.12 — Azimuthal spherical CS

Element

Specification

Description

Azimuthal spherical.

Label

AZIMUTHAL_SPHERICAL

Code

5

Function type

Generating function.

CS descriptor

3D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

none

Coordinate-components

a : azimuth in radians,
q : depression/elevation angle in radians, and
r : radius.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

none

References

[EDM]

 

5.9.7        Lococentric azimuthal spherical CS specification

Table 5.13 — Lococentric azimuthal spherical CS

Element

Specification

Description

Localized azimuthal spherical.

Label

LOCOCENTRIC_AZIMUTHAL_SPHERICAL

Code

6

Function type

Generating function.

CS descriptor

3D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R3, and
      
r, s: axis directions in R3.

Constraints:
      
r and s are orthonormal.

Coordinate-components

a : azimuth in radians,
q : depression/elevation angle in radians, and
r : radius.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

The generating function is the composition of the generating function for azimuthal spherical (see Table 5.12) with the 3D localization operator (see 5.7).

References

[EDM]

 

5.9.8        Geodetic 3D CS specification

Table 5.14 — Geodetic 3D CS

Element

Specification

Description

Geodetic 3D.

Label

GEODETIC

Code

7

Function type

Generating function.

CS descriptor

3D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

a: major semi-axis length
b: minor semi-axis length

Constraints:
      
a > b: (oblate ellipsoid)
      
a = b: (sphere)

Coordinate-components

l : longitude in radians,
f : geodetic latitude in radians, and
h : ellipsoidal height.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

1)       The surface geodetic CS (Table 5.24) is induced on the 3rd coordinate-component surface at any point for which h = 0.

2)       If a = b, the geodetic latitude f  coincides with the spherical latitude q (see Table 5.10).

3)       The inverse generating function is not continuous on the oblate spheroid rotational axis.

References

[HEIK]

 

5.9.9        Planetodetic 3D specification

Table 5.15 — Planetodetic CS

Element

Specification

Description

Planetodetic 3D. Geodetic 3D with longitude in opposite direction.

Label

PLANETODETIC

Code

8

Function type

Generating function.

CS descriptor

3D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

a: major semi-axis length
b: minor semi-axis length

Constraints:
      
a > b: (oblate ellipsoid)
      
a = b: (sphere)

Coordinate-components

f : geodetic latitude in radians,
l : planetodetic longitude in radians, and
h : ellipsoidal height.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

1)       Similar to the geodetic 3D CS (see Table 5.14) except that longitude increases in the opposite direction. In particular, points on a planet surface rotating (prograde) into view have larger planetodetic longitudes than those points rotating out of view. This CS is also called planetocentric when a b and planetographic when a b.

2)       The inverse generating function is not continuous on the oblate spheroid rotational axis.

3)       The coordinate-component ordering differs from that of geodetic 3D CS to satisfy the right handedness requirement.

References

[RIIC]

 

5.9.10    Cylindrical CS specification

Table 5.16 — Cylindrical CS

Element

Specification

Description

Cylindrical.

Label

CYLINDRICAL

Code

9

Function type

Generating function.

CS descriptor

3D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

none

Coordinate-components

formula or figure

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

none

References

[EDM]

 

5.9.11    Lococentric cylindrical CS specification

Table 5.17 — Lococentric cylindrical CS

Element

Specification

Description

Localized cylindrical.

Label

LOCOCENTRIC_CYLINDRICAL

Code

10

Function type

Generating function.

CS descriptor

3D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R3, and
      
r, s: axis directions in R3.

Constraints:
      
r and s are orthonormal.

Coordinate-components

formula or figure

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

See Table 5.16.

Notes

1)       The generating function is the composition of the cylindrical generating function (see Table 5.16) with 3D localization operator (see 5.7).

2)       The domain of the inverse of the generating function is the set of all points not on the axis of the cylinder.

References

[EDM]

 

5.9.12    Mercator CS specification

Table 5.18 — Mercator CS

Element

Specification

Description

Mercator and augmented Mercator map projection coordinate systems.

Label

MERCATOR

Code

11

Function type

Mapping equations.

CS descriptor

Surface (map projection) and 3D (augmented map projection).

Properties

Orthogonal, conformal.

CS parameters and constraints

formula or figure

Coordinate-components

u: easting, and
v: northing.

Augmented coordinate:
       h: ellipsoidal height

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

For f, functional iteration is used for the representation of the inverse mapping equation [SNYD]. Superscripts involving i indicate elements in the iteration sequence.

formula or figure

COM

formula or figure

Point distortion

formula or figure

Figures

formula or figure

Example Mercator map projection

Notes

1)       Meridians project as straight lines that satisfy equations of the form u = a constant. Equally-spaced meridians project to evenly-spaced straight lines orthogonal to the u-axis. Parallels project to straight lines orthogonal to the projected meridians and satisfy equations of the form v = a constant. Evenly-spaced parallels project to unevenly-spaced parallel lines on the projection. The spacing of these lines increases with distance from the u-axis.

2)       The meridian atformula or figure corresponds to the line formula or figure

3)       The point distortion equals formula or figurealong the Equator.

4)       An alternate CS parameter set is given by:

      formula or figure

References

[SNYD]

 

5.9.13    Oblique Mercator spherical CS specification

Table 5.19 — Oblique Mercator spherical CS

Element

Specification

Description

Oblique Mercator and augmented oblique Mercator map projections of a sphere.

Label

OBLIQUE_MERCATOR_SPHERICAL

Code

12

Function type

Mapping equations.

CS descriptor

Surface (map projection) and 3D (augmented map projection).

Properties

Orthogonal, conformal.

CS parameters and constraints

formula or figure

Coordinate-components

u: easting, and
v: northing.

Augmented coordinate:
       h: ellipsoidal height

Domain of the generating function or mapping equations

formula or figure

formula or figure

Generating function or mapping equations

formula or figure

formula or figure

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

formula or figure

Point distortion

formula or figure

Figures

formula or figure

Example oblique Mercator spherical map projection

Notes

1)       The method for specifying the central line by specifying two points on the central line can be accomplished by alternative formulations. The formulations for two such alternatives are provided below.

Alternative a) 

In this alternative the user specifies formula or figure the longitude of one of the two points where the central line crosses the equator and formula or figurethe equator crossing angle at that point. The CS parameters areformula or figure.

The CS parameter constraints are:

formula or figure, and formula or figure

Alternative b) 

In this alternative, a point (l1, f1) on the central line, a central line crossing angle a1 at the point, and k0 at the point are specified. The central line crossing angle is the angle between the central line and the parallel through the given point. The positive sense of the angle is counter-clockwise from east. In this case the origin formula or figureand the equator crossing formula or figure are computed. The CS parameters areformula or figure.

The CS parameter constraints are:

formula or figure,formula or figure , formula or figure

The mathematical formulation is,

formula or figure

2)       Point distortion is equal to k0 at all points on the central line.

References

[SNYD]

 

5.9.14    Transverse Mercator CS specification

Table 5.20 — Transverse Mercator CS

Element

Specification

Description

Transverse Mercator and augmented transverse Mercator map projections.

Label

TRANSVERSE_MERCATOR

Code

13

Function type

Mapping equations.

CS descriptor

Surface (map projection) and 3D (augmented map projection).

Properties

Orthogonal, conformal.

CS parameters and constraints

formula or figure

Coordinate-components

u: easting, and
v: northing.

Augmented coordinate:
       h: ellipsoidal height

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

formula or figure

Point distortion

formula or figure

Simplification in the case of a sphere:

formula or figure

Figures

formula or figure

Example transverse Mercator map projection

Notes

1)       As noted in [SNYD] and [LLEE], an iterative exact solution for the transverse Mercator forward and inverse conversions were developed by Prof. E. H. Thompson in 1945 and were formally published by L. P. Lee in 1962 [LLEE] with the permission of Professor Thompson. In 1980, J. Dozier published a report that adapted Lee’s results to a computerized solution, including complete program listings in the C language [DOZI]. The formulation used here is further adapted by Craig Rollins of the United States National Geospatial-Intelligence Agency (NGA) to include a central scale factor, a non-zero latitude origin and false easting and false northing offsets for both the easting and northing coordinate-components.

2)       The complex functions sn(w |e2), cn(w |e2) and dn(w |e2) are defined in A.8.2.

3)       The CS Generation function is extended by continuity to include the oblate ellipsoid pole points.

4)       The domain of the inverse mapping equations covers the main region of interest. This domain can be extended to a larger region whose exact specification is complicated to define.

5)       The iterative procedures used in both the forward and inverse formulations may be numerically ill conditioned near certain points. This occurs near the boundaries of the domains involved and near the equator for points with negative u values. When implementing these methods in software, special numerical methods may be required in the neighbourhood of such exceptional points. In particular, in the forward conversion, the exceptional points on the equator are avoided by restricting the procedure to use |l*| and then setting u to be negative when l*< 0.

6)       The point distortion equals formula or figurealong the meridian atformula or figure.

References

[DOZI], [LLEE], and [SNYD].

 

5.9.15    Lambert conformal conic CS specification

Table 5.21 — Lambert conformal conic CS

Element

Specification

Description

Lambert conformal conic and augmented Lambert conformal conic map projections.

Label

LAMBERT_CONFORMAL_CONIC

Code

14

Function type

Mapping equations.

CS descriptor

Surface (map projection) and 3D (augmented map projection).

Properties

Orthogonal, conformal.

CS parameters and constraints

formula or figure

Coordinate-components

u: easting, and
v: northing.

Augmented coordinate:
       h: ellipsoidal height

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

formula or figure

Point distortion

formula or figure

Figures

formula or figure

Example Lambert conformal conic map projection

Notes

1)       formula or figureis the signum function (see ISO 31-11:1992).

2)       The surface geodetic coordinate formula or figureprojects to map coordinate formula or figure

3)       The point distortion is unity along the standard parallel(s) f1 and f2.

4)       An alternate CS parameter set is given by: formula or figure

References

[SNYD]

 

5.9.16    Polar stereographic CS specification

Table 5.22 — Polar stereographic CS

Element

Specification

Description

Polar stereographic and augmented polar stereographic map projections.

Label

POLAR_STEREOGRAPHIC

Code

15

Function type

Mapping equations.

CS descriptor

Surface (map projection) and 3D (augmented map projection).

Properties

Orthogonal, conformal.

CS parameters and constraints

formula or figure

Coordinate-components

u: easting, and
v: northing.

Augmented coordinate:
       h: ellipsoidal height

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

For f, functional iteration is used for the representation of the inverse mapping equation. Superscripts involving m indicate elements in the iteration sequence.

formula or figure

COM

formula or figure

Point distortion

formula or figure

Figures

formula or figure

Example polar stereographic map projection

See also Figure 5.7

Notes

1)       Meridians project as straight lines radiating from the point formula or figure. Parallels project to concentric circles.

2)       The point distortion values at pole is: formula or figure.

3)       An alternate CS parameter set is given by: formula or figure

4)       In the case of a sphere, the mapping equations are derived from the stereographic projection (see Figure 5.7).

5)       The CS Generation function is extended by continuity to include the oblate ellipsoid pole points.

References

[SNYD]

 

5.9.17    Equidistant cylindrical CS specification

Table 5.23 — Equidistant cylindrical CS

Element

Specification

Description

Equidistant cylindrical and augmented equidistant cylindrical map projections.

Label

EQUIDISTANT_CYLINDRICAL

Code

16

Function type

Mapping equations.

CS descriptor

Surface (map projection) and 3D (augmented map projection).

Properties

Orthogonal, non-conformal.

CS parameters and constraints

formula or figure

Coordinate-components

u: easting, and
v: northing.

Augmented coordinate:
       h: ellipsoidal height

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

formula or figure

Point distortion

formula or figure

Figures

formula or figure

Example equidistant cylindrical map projection

Notes

1)       Meridians project as straight lines that satisfy equations of the form u = a constant. Equally-spaced meridians project to evenly-spaced straight lines orthogonal to the u-axis. Parallels project to straight lines orthogonal to the projected meridians and satisfy equations of the form v = (a constant).

2)       formula or figure formula or figure indicates true scale, or "equidistance" along meridians.

3)       The radius R of the conceptual cylinder is R = a k0.

4)       An alternate CS parameter set is given by:

       formula or figure

References

[SNYD]

 

5.9.18    Surface geodetic CS specification

Table 5.24 — Surface geodetic CS

Element

Specification

Description

Surface geodetic.

Label

SURFACE_GEODETIC

Code

17

Function type

Generating function.

CS descriptor

Surface curvilinear.

Properties

Orthogonal.

CS parameters and constraints

a: major semi-axis length
b: minor semi-axis length

Constraints:
      
a > b: (oblate ellipsoid)
      
a = b: (sphere)

Coordinate-components

l : longitude in radians, and
f : geodetic latitude in radians.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

1)       The CS surface is the oblate ellipsoid (or sphere) surface excluding the pole points.

2)       The geodetic 3D CS (Table 5.14) induces this CS on the 3rd coordinate-component surface at any point for which h = 0.

3)       If a = b, the geodetic latitude f  coincides with the spherical latitude q (see Table 5.10).

4)       The inverse generating function is not continuous at the pole points formula or figure

References

[HEIK]

 

5.9.19    Surface planetodetic CS specification

Table 5.25 — Surface planetodetic CS

Element

Specification

Description

Surface planetodetic. Surface geodetic with longitude in opposite direction.

Label

SURFACE_PLANETODETIC

Code

18

Function type

Generating function.

CS descriptor

Surface curvilinear.

Properties

Orthogonal.

CS parameters and constraints

a: major semi-axis length
b: minor semi-axis length

Constraints:
      
a > b: (oblate ellipsoid)
      
a = b: (sphere)

Coordinate-components

f : geodetic latitude in radians, and
l : planetodetic longitude in radians.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

1)       Similar to surface geodetic CS (see Table 5.24) except that longitude is in the opposite direction. In particular, points on a planet surface rotating (prograde) into view have larger planetodetic longitudes than those points rotating out of view.

2)       The inverse generating function is not continuous at the pole points formula or figure

3)       The coordinate-components are ordered for compatibility with planetodetic 3D CS (see Table 5.15)

References

[RIIC]

 

5.9.20    Lococentric surface Euclidean CS specification

Table 5.26 — Lococentric surface Euclidean CS

Element

Specification

Description

Localization of Euclidean 2D CS into plane surface in 3D position-space.

Label

LOCOCENTRIC_SURFACE_EUCLIDEAN

Code

19

Function type

Generating function.

CS descriptor

Surface linear.

Properties

Orthonormal.

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R3, and
      
r, s: axis directions in R3.

Constraints:
      
r and s are orthonormal.

Coordinate-components

u, v

Domain of the generating function or mapping equations

R2

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

1)       The CS surface is the plane specified by:formula or figure.

2)       The generating function is the composition of the generating function for Euclidean 2D (see Table 5.29) with the surface localization operator (see 5.7).

References

[EDM]

 

5.9.21    Lococentric surface azimuthal CS specification

Table 5.27 — Lococentric surface azimuthal CS

Element

Specification

Description

Localization of azimuthal CS into plane surface in 3D position-space.

Label

LOCOCENTRIC_SURFACE_AZIMUTHAL

Code

20

Function type

Generating function.

CS descriptor

Surface curvilinear.

Properties

Orthogonal.

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R3, and
      
r, s: axis directions in R3.

Constraints:
      
r and s are orthonormal.

Coordinate-components

a : azimuth in radians, and
r : radius.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

1)       The CS surface is the plane specified by:formula or figure.

2)       The generating function is the composition of the generating function for azimuthal CS (see Table 5.31) with the 3D localization operator (see 5.7).

References

[EDM]

 

5.9.22    Lococentric surface polar CS specification

Table 5.28 — Lococentric surface polar CS

Element

Specification

Description

Localization of polar CS into plane surface in 3D position-space.

Label

LOCOCENTRIC_SURFACE_POLAR

Code

21

Function type

Generating function.

CS descriptor

Surface curvilinear.

Properties

Orthogonal.

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R3, and
      
r, s: axis directions in R3.

Constraints:
      
r and s are orthonormal.

Coordinate-components

formula or figure

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

1)       The CS surface is the plane specified by:formula or figure.

2)       The generating function is the composition of the generating function for polar CS (see Table 5.33) with the surface localization operator (see 5.7).

References

[EDM]

 

5.9.23    Euclidean 2D CS specification

Table 5.29 — Euclidean 2D CS

Element

Specification

Description

Euclidean 2D.

Label

EUCLIDEAN_2D

Code

22

Function type

Generating function.

CS descriptor

2D linear.

Properties

Orthonormal.

CS parameters and constraints

none

Coordinate-components

u, v

Domain of the generating function or mapping equations

R2

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

R2

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

Coordinate-space 2-tuples are identified with position-space 2-tuples.

References

[EDM]

 

5.9.24    Lococentric Euclidean 2D CS specification

Table 5.30 — Lococentric Euclidean 2D CS

Element

Specification

Description

Localized Euclidean 2D

Label

LOCOCENTRIC_EUCLIDEAN_2D

Code

23

Function type

Generating function.

CS descriptor

2D linear.

Properties

Orthonormal.

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R2, and
      
r, s: axis directions in R2.

Constraints:
      
r and s are orthonormal.

Coordinate-components

u, v

Domain of the generating function or mapping equations

R2

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

R2

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

1)       Euclidean 2D CS is a special case withformula or figure.

2)       The generating function is the composition of the generating function for Euclidean 2D CS (see Table 5.29) with the 2D localization operator (see 5.7).

References

[EDM]

 

5.9.25    Azimuthal CS specification

Table 5.31 — Azimuthal CS

Element

Specification

Description

Azimuthal coordinate system.

Label

AZIMUTHAL

Code

24

Function type

Generating function.

CS descriptor

2D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

none

Coordinate-components

a : azimuth in radians, and
r : radius.

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

none

References

[EDM]

 

5.9.26    Lococentric azimuthal CS specification

Table 5.32 — Lococentric azimuthal CS

Element

Specification

Description

Localization of azimuthal CS.

Label

LOCOCENTRIC_AZIMUTHAL

Code

25

Function type

Generating function.

CS descriptor

2D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R2, and
      
r, s: axis directions in R2.

Constraints:
      
r and s are orthonormal.

Coordinate-components

formula or figure

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

The generating function is the composition of the generating function for azimuthal CS (see Table 5.31) with the 2D localization operator (see 5.7).

References

[EDM]

 

5.9.27    Polar CS specification

Table 5.33 — Polar CS

Element

Specification

Description

Polar coordinate system.

Label

POLAR

Code

26

Function type

Generating function.

CS descriptor

2D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

none

Coordinate-components

formula or figure

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

none

References

[EDM]

 

5.9.28    Lococentric polar CS specification

Table 5.34 — Lococentric polar CS

Element

Specification

Description

Localized polar.

Label

LOCOCENTRIC_POLAR

Code

27

Function type

Generating function.

CS descriptor

2D curvilinear.

Properties

Orthogonal.

CS parameters and constraints

Localization parameters:

       q: the lococentric origin in R2, and
      
r, s: axis directions in R2.

Constraints:
      
r and s are orthonormal.

Coordinate-components

formula or figure

Domain of the generating function or mapping equations

formula or figure

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

formula or figure

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

formula or figure

Notes

The generating function is the composition of the generating function for the polar CS (see Table 5.33) with the 2D localization operator (see 5.7).

References

[EDM]

 

5.9.29    Euclidean 1D CS specification

Table 5.35Euclidean 1D CS

Element

Specification

Description

Euclidean 1D.

Label

EUCLIDEAN_1D

Code

28

Function type

Generating function.

CS descriptor

1D linear

Properties

none

CS parameters and constraints

none

Coordinate-components

u

Domain of the generating function or mapping equations

R1

Generating function or mapping equations

formula or figure

Domain of the inverse of the generating function or mapping equations

R1

Inverse of the generating function or mapping equations

formula or figure

COM

n/a

Point distortion

n/a

Figures

none

Notes

Coordinate-space 1-tuples are identified with position-space 1-tuples.

References

[EDM]



[ Table of contents ]



5 The generating function properties and the implicit function theorem together imply that for each point in the interior of the CS domain, there is an open neighbourhood of the point whose image under the generating function lies in a smooth surface. This requirement specifies that there exists one smooth surface for all of the points in the CS domain. This requirement is specified to exclude mathematically pathological cases.

6 The generating function properties and the implicit function theorem together imply that for each point in the interior of the CS domain, there is an open neighbourhood of the point whose image under the generating function lies in a smooth curve. This requirement specifies that there exists one implicitly-defined smooth curve for all the points in the CS domain. This requirement is specified to exclude mathematically pathological cases.

7 The ISO 19111 term for this concept is “coordinates”.

8 The ISO 19111 term for this concept is “coordinate”.

9 The ISO 19111 concept of a linear reference system is a specialization of the curve CS and plane curve CS concepts.

10 ISO 19111 defines the term “meridian” as the intersection between an ellipsoid and a plane containing the semi-minor axis of the ellipsoid.

11 ISO 19111 defines the term “prime meridian” as the meridian from which the longitudes of other meridians are quantified. In Clause 7, most, but not all, oblate ellipsoid Earth object reference models associate the Greenwich meridian with the prime meridian (see 7.4.5).  

12 Some publications use “rectangular” to denote an orthogonal linear CS, and “oblique” to denote a non-orthogonal linear CS.

13 ISO 19111 defines “Cartesian coordinate system” as a coordinate system that gives the position of points relative to n mutually-perpendicular axes.

14 It is a consequence of the Theorema Egregium of Gauss that no map projection CS can eliminate all distortion.

15 This concept is found in the literature under a variety of names.  The term “point distortion” is introduced to avoid ambiguity.

16 The distinction between a map projection coordinate and a display coordinate is not usually made explicit in the literature. The term “display coordinate” is introduced to avoid ambiguity.

17 More general definitions that allow measurements of azimuth angle clockwise or counter-clockwise and from the north or south side of the meridian are in use. The generalization to the case for which one or more of the two points is not on the surface is treated in [RAPP1] and [RAPP2]. The more general definitions are not required for subsequent SRM concepts.

18 The function arctan2 is defined in A.8.2.