[ 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 k^th-component of a coordinate n-tuple (1 ≤ k ≤ n) may be called the k^th coordinate-component. Coordinate-component8 is the collective term for any k^th 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:

where:

The CS domain of F in coordinate-space is .

The CS range of F in position-space is .

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.

Figure 5.1 — Polar CS geometry

Figure 5.2 — The 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

 

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 = (u₀, v₀, w₀) is in the interior of the CS domain D, then three surface CS generating functions at u are defined by:

              ,

              , and

              .

The CS domain for S₁ is the connected component of which contains (v₀, w₀).
The CS domain for S₂ is the connected component of which contains (u₀, w₀).
The CS domain for S₃ is the connected component of which contains (u₀, v₀).

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

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

EXAMPLE 1         Coordinate-component surface: The geodetic 3D CS with generating function is specified in Table 5.14 with CS parameters a and b. The 3^rd coordinate-component surface at is 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 3^rd surface CS induced by the geodetic 3D generating function at . 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 = (u₀, v₀, w₀) is in the interior of D, then the 1^st, 2^nd, and 3^rd coordinate-component curves at u are parametrically specified, respectively, by the following smooth functions:

              ,

              , and

              .

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

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

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 = (u₀, v₀) is in the interior of D, then the 1^st and 2^nd coordinate-component curve at u is parametrically specified, respectively, by the following smooth functions:

              , and

              .

The domain for C₁ is the connected component of which contains u₀.
The domain for C₂ is the connected component of which contains v₀.

EXAMPLE            If is 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 is, and the 2^nd coordinate-component curve is.

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

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

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

The meridian at shall be called the prime meridian11.

The parallel at shall be called the equator.

Figure 5.3 — Geodetic 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-space.

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 k^th coordinate-component curve is a line. The k^th coordinate-component curve at the origin 0 of a linear CS is the k^th-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  (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  in the interior of the CS domain, the coordinate-component curves at p determine an ordered set of three tangent vectors:

An orthogonal CS of CS type 3D is a right-handed CS if for some coordinate  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  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  is specified to satisfy the right-handed CS requirement.

EXAMPLE 2         The surface planetodetic geodetic CS (Table 5.25)) coordinate-component ordering  is determined by the coordinate-component ordering  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

 

Table 5.3 — Localization inverse operators

 

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 localization operator. The localized CS is of the same CS type (CS type 3D or CS type surface, respectively). Its generating function is  and has the same CS domain as F.

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

a.       A 2D CS with generating function F_C is localized by composing F with the localization operator. The localized CS is a surface CS. Its generating function is  and has the same CS domain as F.

b.       A 2D CS with generating function F is localized by composing F with the localization operator. The localized CS is a surface CS. Its generating function is  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

 

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  and  of the generating projection P shall be called the mapping equations:

             

where:

             

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  and  of the inverse generating projection:

             

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 function, the resulting function  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  are not in the MP domain of P nor are they in the range of Q. However, if the composite function is continuous at the pole points, then and shall be extended by continuity to include the pole points in the CS range.

NOTE          The CS generating function is not to be confused with the generating projection P.

 

Figure 5.4 — The 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  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 confo