Spatial Reference Model

Annex A

(Normative)

Mathematical foundations

A.1     Introduction and table of contents

A.1.1       Table of contents

Table A.1-1 -- Table of contents

 

Table A.1-2 -- Table of tables

 

Table A.1-3 -- Table of figures

A.1.2       Overview

{ensure a consistent underpinning for concepts developed/used in this IS}

A.2     Rⁿ as a real vector space

A.2.1       Properties

An n-tuple of real numbers, a, is an ordered set of real numbers, denoted by a = (a₁, a₂, a₃, , a_n). The space of all n-tuples of real numbers, denoted by Rⁿ, is a vector space with the canonical basis, , , .   The elements of Rⁿ are variously called points or vectors (the latter term is used in the context of directions or vector space operations).[problem 4]  The following definitions apply to any vectors , and  in Rⁿ: [prob. 1:]

a.       The inner product of two vectors x and y (also called a dot-product), is defined as:

        .

b.       Two vectors x and y, are orthogonal if = 0.  x is orthogonal to a set of vectors, if x is orthogonal to each vector belonging to the set. [prob 3: add definition]

Note that when n = 2 or 3, , where  is the angle between x and y. Two such vectors are perpendicular if and only if they are orthogonal.

c.       A norm || x || derived from the inner product defined by .

Note that only the zero vector 0 = (0, 0, ) has norm zero.

d.       x is called normalized if . [prob. 2 fix: name change: normal -> normalized.  Reserve normal for surface normal]

e.       A set of two or more normalized and pair-wise orthogonal vectors is an orthonormal set of vectors.

f.        A metric D, called the Euclidian metric, defined by. The value of D is called the Euclidiandistance between x and y.

g.       The cross product of two vectors x and y in R³ is a vector, defined as:
.
Note that x y is orthogonal to both x and y, and , where  is the angle between vectors x and y.

[prob. 5: last paragraph removed]

A.2.2       Smooth functions on Rⁿ

A real valued function, f, defined on an open domain in Rⁿ is called smooth if its partial derivatives of all orders exist at each point in its domain. The vector of first order partial derivatives

is called the gradient of f.

A vector valued function, F, with an open domain, D, in Rⁿ and range in R^m is called smooth if each component function, f_i, is smooth where the component function f_i is the real valued function representing the i^th component of the vector valued function F and is defined by f_i(v) = e_iF(v) where e_i is the i^th canonical basis vector. In this case:

F(v) = (f₁(v), f₂(v), f₃(v), , f_m(v)) for v = (v₁, v₂, v₃, ,v_n) in D.

The n  m matrix of partial derivatives evaluated at a point in the domain D:

is called the first derivative of F, and is denoted dF. The first derivative is also called the Jacobian matrix at the point v. In the case of m = n, the Jacobian matrix is square, and its determinant is called the Jacobian determinant. A smooth function Rⁿ valued function defined on a domain, D, in Rⁿ is said to be orientation preserving if its Jacobian determinant is strictly positive for all points in D.

A vector value function F defined on all of Rⁿ is called linear if:

            F(ax + y) = aF(x) + F(y) for all real scalars, a, and vectors x and y in Rⁿ.

A.3     Smooth surfaces in R³

A.3.1       Definition and examples

A smooth surface in R³ is implicitly defined by a real valued smooth function F on R³ as the set, S, of all points, (x, y, z), in R³ satisfying:

a.       F(x, y, z) = 0, and

b.       grad(F)(x, y, z) (0, 0, 0).

In which case F is called a surface generating function for the surface S.

The implicit function theorem implies that tangent planes (and normal vectors) are defined at each (x, y, z) in the surface S.

Example 1:

An oblate spheroid (OBS) with major axis a₁ and minor axis a₂, where 0 < a₂  a₁, is generated by the function

.

Note that F is smooth and that

is never (0, 0, 0) on the OBS surface F = 0.

Note: The special case of a₂ = a₁ = r is called a sphere of radius r. An equivalent generating function is given by .

Example 2:

Let n be a non-zero vector in R³, and define F(v) = vn for all v = (u, v, w) in R³ then the surface generated by F = 0 is a plane through the origin perpendicular to n.

Note that F is smooth and .

Special cases:

When n = (1, 0, 0), F(u, v, w) = u generates the VW-plane.

When n = (0, 1, 0), F(u, v, w) = v generates the UW-plane.

When n = (0, 0, 1), F(u, v, w) = w generates the UV-plane.

Example 3:

Let p be a point on a surface S with generating function G, and let n  = grad(G)(p). Then if Fis defined as F(v) = n(v - p), the surfacegenerated by F = 0 is called the tangent plane to the surface S at point p.

A.3.2       Smooth curves in Rⁿ

A smooth curve in Rⁿ is parametrically defined as a smooth one-to-one Rⁿvalued function, F(t), defined on an interval, I, in R such that ||dF(t)||0 for any t in I.The direction of the first derivative vector dF =  (df₁/dt, df₂/dt, , df_n/dt) evaluated at t₀ is tangent to the curve at the point p = F(t₀) (see Example 4).

Closed curves: If a smooth function F is defined on a closed and bounded interval I with interval end points t₀ and t₁, and if F defines a smooth curve on the interior of I and p=F(t₀)=F(t₁), then F is said to generate a closed curve through p.

A smooth curve C in R³ (defined by F on the interval I) is said to lie in the surface S generated by a function g, if g(F(t)) = 0 for all t in I. In this case, we say that C is a surface curve in S. A smooth surface S is said to be connected if given any two (distinct) points in S, there exits a smooth curve that contains the two points and lies in the surface.

A smooth curve in R² may be implicitly defined by a real valued smooth function F on R² as the set, S, of all points, (x, y), in R² satisfying:

c.       F(x, y) = 0, and

d.       grad(F)(x, y) (0, 0).

If p=F(t_p) and q=F(t_q) are two points on a smooth curve defined by F, and t_p < t_q then the arc length from p to q along the curve is defined by

            (integrated between t_p and t_q).

Example 4:

Let p and n be vectors in Rⁿ such that n is not the zero vector in Rⁿ), and then a line that passes through p in direction n is parametrically defined by the smooth curve

L(t) = p + t n, - < t < +.   (Note that ||dL(t)|| = ||n|| > 0.)

If C(t) is a smooth curve C passing through a point p = C(t_p) and n = dC(t_p), then T(t) = p + t n, - < t < + defines a line called the tangent line to the curve C at p.

Note that if C(t) defines a surface curve in S with generating function G which passes through p = C(t_p), then the tangent line to the curve at p, T(t) = p + t dC(t_p), lies in the tangent plane to the surface S at p (defined by F = 0 in Example 2a).  (Since G(C(t)) = 0, the chain rule implies that  grad(G)  dC = d(G(C(t)))/dt = 0, so that F(T(t)) = F(p  + t  dC(t_p)) = t(grad(G) dC) =  0.)

Example 5:

An ellipse in R² with major axis a₁ and minor axis a₂, where 0 < a₂ a₁, is parametrically defined as

F(t) = (a₁cos(t), a₂sin(t)), for all  in the interval - < t .

Note that that F is smooth and ||dF(t)||  a₂ > 0 for all t in the interval.

Example 6:

An ellipse in R² with major axis a₁ and minor axis a₂, where 0 < a₂ a₁, is implicitly generated by the real valued function

.

Note that F is smooth and that

is never (0, 0) on the surface F = 0.

A.3.3       Surface normals and orientable surfaces

If p is a point on a smooth surface S, then a normal vector nof S at p is defined as any normalized vector which is orthogonal to the tangent plane of S at p (see example 3). There are two possible directions of a normal n at p. One of these can be selected and called the positive normal direction of S at p. {need to clarify terminology using “unit normal vector”}

A connected smooth surface S is called orientable if the positive normal direction of an arbitrary point p on S can be continued in a unique and continuous manner to the entire surface. A normal direction at a fixed point p₀ may be continued if there does not exist a closed curve C on S through p₀ such that the positive normal direction reverses when it is displaced continuously from p₀ along C and back to p_0. A Mobius strip is an example of a non-orientable surface.

http://www.sedris.org/Specifications/SRM/index.html

 

 

9.6.4.2.3.2    Transformation to coordinates based on a local datum

In certain cases, x, y, and z ellipsoid of revolution transformation parameters can be applied directly to convert celestiocentric coordinates based on the WGS 84 global datum to celestiocentric coordinates based on a local datum, as follows:

.

. . . with , , , and s set to zero.

 

 

 

 

 

5.3.3.2    Curvilinear 3D Coordinate Systems

Let D be an connected open domain in R³ with possibly some boundary points, and let

x = X(u₁, u₂, u₃); {notation consistency?}

y = Y(u₁, u₂, u₃);

z = Z(u₁, u₂, u₃);

be smooth (non-linear) functions on D such that the Jacobian (x, y, z)/  (u₁, u₂, u₃) is non zero in D. Then if G: (u₁, u₂, u₃)  (x, y, z) is one-to-one, it is a generating function for a 3D coordinate system for the image of D, and (u₁, u₂, u₃) are called curvilinear coordinates for points in the image.

Example 1:

Cylindrical coordinates (r, , z) where, and z = z with domain D = {(r, , z) | r  > 0, -    < , - < z  < }.

Example 2:

Spherical coordinates (r, , ) where with domain D = {(r, , ) | r  > 0, -    < , -/2 <    < /2}. The image of D is all of R³ except for the Z-axis. The Spherical Coordinate System is illustrated in Figure 5.3.2. [Caution: In some publications, the angle  is measured from the positive Z-axis rather that from the XY-plane.]

 

5.4.1.1    Definition

A projection is a Surface Coordinate System (see Error! Reference source not found.) that is defined by a smooth 1-1 function that associates a subset of points on a smooth surface with a set of points (called the range of the projection) in R². This function is called the generating projection.

The classical Mercator projection for an OBS with eccentricity e{just use a sphere} provides an example. The generating projection is given by M: (,)  (x, y) = (f(,), g(,)), where

The domain of M is {(,) | - < -_origin    and -/2 <  < /2} and the range is {(x, y)  R² | -a < x    a}.