HOMOGENEOUS TRANSFORMATION MATRICES

Daniel W. VanArsdale

Vector (nonhomogeneous) methods are still being recommended to effect rotations and other linear transformations. Homogeneous matrices have the following advantages:

•  simple explicit expressions exist for many familiar transformations including rotation
•  these expressions are n-dimensional
•  there is no need for auxiliary transformations, as in vector methods for rotation
•  more general transformations can be represented (e.g. projections, translations)
•  directions (ideal points) can be used as parameters of the transformation, or as inputs
•  if nonsingular matrix T transforms point P by PT, then hyperplane h is transformed by T-1h
•  the columns of T (as hyperplanes) generate the null space of T by intersections
•  many homogeneous transformation matrices display the duality between invariant axes and centers.

The expressions below use reduction to echelon form and Gram-Schmidt orthonormalization, both with slight modifications. They can be easily coded in any higher level language so that the same procedures generate transformations for any dimension. This article is at an undergraduate level, but the reader should have had some exposure to linear algebra and analytic projective geometry. This material is based on: Daniel VanArsdale, Homogeneous Transformation Matrices for Computer Graphics, Computers & Graphics, vol. 18, no. 2, pp. 177-191, 1994.  Some references are given at the end; authors' names within the text are clickable to these. A few annotated links to projective geometry sites are also listed.

The intent of this document is to provide "cook book" information  to understand, code or use the homogeneous transformation matrices presented. Derivations, proofs, and additional results appear elsewhere in a companion site: "Homogeneous Coordinates: Methods."  Before reading Methods one should first browse through the conventions and procedures of sections I and II below.

CONTENTS

I.      Definitions and Notation
A. General   B. Points  C. Hyperplanes   D. Flats   E. Point Matrices   F. Hyperplane matrices   G. Projective transformations

II.    Three Procedures
A. Normal  B. Oriented hyperplane representation  C. Orthonormalization

III.   Transformation Matrices
A. Projection   B. General Collineation   C. Affinity   D. Isometry   E. Translation   F. Dilation & Reflection   G. Strain & Shear   H. Rotation

I. DEFINITIONS AND NOTATION

A. GENERAL

Rn the real projective space of dimension n - 1.
Rn  =  n-tuples of real numbers, not all zero.
n = the rank of the space, the dimension plus 1.

I = the n x n identity matrix.   Is = the s x s identity matrix.
0 = a matrix of all zeros of appropriate size.
M-1 = the inverse of square matrix M.  det M = the determinant of M.
rank(A) = the rank of a matrix A.

A semicolon between elements of a matrix designates vertical stacking - begin the following element on the next line. Thus [a,b,c; d,e,f] is the 2 x 3 matrix with first row  [a,b,c] and second row [d,e,f].

/= means not equal.  <= means less than or equal
S1 c S2 means set S1 is contained in, or equal to, set S2.

B. POINTS

P = [X1, X2, ..., Xn] = the homogeneous coordinates of point P, a 1x n row matrix, P / = 0.
For any nonzero constant c, both P and cP =  [cX1, cX2, . . ., cXn] represent the same point.
The "point" is the class of all such representations, but for convenience we may identify a particular  representation as the point.

X1 is the homogeneous coordinate of point P. Often in the literature the homogeneous coordinate of a point is placed last (Xn) instead.

If X/= 0, P is ordinary and corresponds to the Cartesian point (X2/X1, . . ., Xn/X1).
P (ordinary) is normalized if X1 = 1.

If X1 = 0,  P is ideal and may represent  the direction of the Cartesian vector (X2, . . ., Xn). In a projective context P is the same ideal point as -P =  (0, -X2, . . ., -Xn), but in practice we often distinguish these directions.
P (ideal) is normalized if  (X2)2 + . . . + (Xn)2 = 1.

Example: In R3 homogeneous coordinates [3, 2, 1] represent the same ordinary point P as coordinates [1, 2/3, 1/3], the normalized form of P.  P corresponds to the point in the plane with Cartesian coordinates (2/3, 1/3). In R3 the ideal point V = [0, 3/5, 4/5] is normalized since (3/5)2 + (4/5)2 = 1. V may be used to represent the direction of the planar vector (3/5, 4/5).

C. HYPERPLANES

h = [Y1;Y2; . . .; Yn] =  the coordinates of a hyperplane h, an n x 1 column matrix, h /= 0.
For any nonzero constant c, both h and ch =  [cY1; cY2; . . .; cYn]  represent the same hyperplane. The "hyperplane" is the class of all such representations, but for convenience we may identify a particular representation as the hyperplane.
Hyperplane h contains the points Q for which Qh = 0.  These points are the null space of h, and may be designated as null(h) or hP.

For h as above, let C = Y22+. . . + Yn2:
If C = 0 h is ideal, the unique hyperplane at infinity represented by w = [1; 0; . . .; 0].
If C /= 0 h is ordinary, and normalized if C = 1. Normalize any ordinary hyperplane h by dividing its coordinates by the positive square root of C.

Example: In R4 (three dimensional space) the 4 x 1 column matrix h = [1; -1: -1; -1] represents the ordinary plane through the points X = [1,1,0,0], Y=[1,0,1,0] and Z = [1,0,0,1] since Xh = Yh = Zh = 0. Hyperplane h can be normalized by dividing each component by the square root of 3.

D. FLATS

S = a flat of rank r = the set of all points of the form c1P1 + . . .+ crPr, the ci any constants (not all zero), the Pi some r <= n fixed independent points.  As in linear algebra, the P are called a basis of the flat S.  The rank of S is designated rank(S).

Flats need not contain the origin [1,0, . . ., 0].
The null set of points is considered a flat of rank zero.  Rn is a flat of rank n. These two flats are improper, all others are proper.
Flat S is ordinary if it contains an ordinary point. An ideal flat contains an ideal point and no ordinary points.

S1 ^ S2 = the intersection (meet) of flats S1 and S2 , also a flat.
S1v S2 = the union (join) of flats S1 and S2 , also a flat.
rank(S1) + rank(S2) = rank(S1v S2) + rank(S1 ^ S2)    (Ayres, p. 87)

In Rn  flats S1 and S2 are complementary if they are disjoint and rank(S1) + rank(S2) = n.

E. POINT MATRICES

P = a point matrix = any r x n matrix, P = [P1; P2; . . .; Pr], r >= 1. The 1 x n matrices Pi are the rows of P, and if nonzero represent points in Rn.  Point matrices are represented  by upper case letters, or by upper case superscripts.

P represents the flat spanned by its rows  P1, P2, . . ., Pr. This flat is also called the row space of  P, or the union of  P1, P2, . . ., Pr.  It may be designated as range(P), or in some contexts simply by the matrix P itself.

P is ordinary if range(P) is ordinary.
P is independent if its rows P1, P2, . . . , Pr are independent.

Say P1, . . ., Pn are n ordered independent points in Rn and let P = [P1; . . .; Pn]. Then P1, . . ., Pn have orientation the sign of det P.  This matches the usual convention for positive orientation, and thus motivates placement of the homogeneous coordinate first instead of last.

F. HYPERPLANE MATRICES

h = a hyperplane matrix = any n x s matrix, h = [h1, h2, . . . , hs], s <= n. The n x 1 matrices hi are the columns of h, and if nonzero represent hyperplanes. Hyperplane matrices are represented by lower case letters, or by lower case superscripts.

h represents the null space of h, all points P for which Ph = 0. This flat is the intersection (meet) of the  hyperplanes h1, . . ., hs and may be designated by null(h) or hP.

h is ordinary if hP is ordinary.
h is independent if its columns h1, h2, . . ., hs are independent.
If the rank of independent hyperplane matrix h in Rn is s, the rank of the flat it represents by intersections is n - s.

Example: In R4 the x-axis can be represented by the point matrix P = [1, 0, 0, 0; 1, 1, 0, 0] or
by the hyperplane matrix h = [h1, h2] where h1 = [0; 0; 1; 0] and h2 = [0; 0; 0; 1].

G. PROJECTIVE TRANSFORMATIONS

A projective transformation f on Rn (real projective space of dimension n-1) is a mapping of a set of points of Rn (the domain of f) to Rn for which there exists an n x n matrix T of real numbers such that:
(1) for all points P in the domain of  f,  Pf is represented by PT,
(2) for all points P not in the domain of  f, PT = 0.
Matrix T then represents the transformation  f, as does cT, c any nonzero constant.
For convenience we will often identify a projective transformation with a matrix that represents it.
Alternatively, if PT = 0 we can regard P as mapped to the null flat by T. Then any projective transformation maps flats to flats.

If T is nonsingular it represents a one-to-one mapping of  Rn onto  Rn called a collineation.
For collineation T,  hyperplane h is mapped to hyperplane T-1 h.
Collineations map straight lines to straight lines.

For a projective transformation f on Rn represented by matrix T:
The range of  f is the set of all image points under f, designated by range(f) or range(T).
Range(T) is a flat with rank equal to rank(T).
The null space of f is the set of all points of Rn that are not in the domain of  f, which is all points P such that PT = 0.  These points are designated by null(f), null(T), or  TP.
For any T,  rank[T] + rank[null(T)] = n

II. THREE PROCEDURES

A. The NORMAL, hN, of a hyperplane matrix h.

In three dimensional vector space points (x, y, z) on a plane satisfy an equation of the form  Ax + By + Cz  + D = 0,  (A, B, C, D constants not all zero). A normal (perpendicular) vector to this plane is  (A, B, C). Using homogeneous coordinates the plane is represented by the column matrix h = [D; A; B; C] and points P = [k, x, y, z] on the plane satisfy Ph = 0. The normal to plane h is the ideal point represented by the row matrix [0, A, B, C], which we designate hN.  For a hyperplane matrix g with more than one column, say g = [h1, h2], we define gN = [ h1N; h2N ], an ideal point matrix with two rows. When g is "orthonormalized" (see Procedure C) then gN g = I2. Normals are used in several of the transformation matrices below.

Procedure A: Find the normal, hN, of a hyperplane matrix h.
Step 1. Transpose h.
Step 2. Set the first (homogeneous) column of the transpose to zero.

It can be shown that for any hyperplane matrix h, h and hN have the same rank if and only if h is ordinary. The normal of the hyperplane at infinity, w, is undefined.  A "normal" is an ideal point or flat, whereas "normalization" is the unit scaling of homogenous coordinate representations of points, hyperplanes, and matrices.

B. The ORIENTED HYPERPLANE REPRESENTATION, Ph.

A flat S in Rn may be designated by r independent points that generate S by their union, or by n - r independent hyperplanes that form S by their intersection. Usually it is easier to visualize a flat as the union of points. For example, in three dimensional space, the axis line of a rotation could be designated by two points it contains, and the invariant plane of a reflection may be designated by three points on the plane.  But a hyperplane representation of a flat is very useful, and appears in most of the transformation matrices below. Thus we need a procedure to convert from a point matrix representation P to a hyperplane representation g, written g = Ph. This can be done by solving a system of homogeneous linear equations, as by elementary column operations.

An additional condition may be imposed on g. If the axis of a rotation is designated as the line through points P1 and P2 this implies a sense of rotation opposite that if the two points are taken in the order P2 and P1. In converting this line to a hyperplane representation h = [h1, h2] we need to assure that the orientation of the four points P1, P2,  h1N, and h2N, in that order, is positive. This can be done by tallying a parity during the elementary column operations ( VanArsdale ).

Procedure B: Given an r x n independent point matrix P representing flat S by unions, find an independent hyperplane matrix g = Ph such that:  (1) g represents S by intersections, and (2) if P is ordinary, det [P; gN] > 0.

Step 1. Form the (r+n) x n compound matrix [P; I]. Set variable sgn to 1.

Step 2. Reduce [P; I] by elementary column operations to matrix [Q; E] so that (i) the first r columns of Q form a lower triangular matrix with ones on the diagonal, (ii) the remaining n - r columns of Q contain all zeroes. In this reduction, whenever two columns are interchanged or a column is multiplied by a negative number, set sgn = - sgn.

Step 3. Set g = Ph to the last n - r columns of E. Multiply the first column of g by sgn.

With two easy modifications Procedure B can be used to find general intersections (see  Methods).  An example using Procedure B appears below (IV- A).

C. ORTHONORMALIZATION of an ordinary hyperplane matrix.

In vector analysis the independent vectors v1, v2, . . ., vs span an s-dimensional subspace, S, by linear combinations. The familiar Gram-Schmidt orthogonalization process uses linear combinations of the vi to produce vectors V1, V2, . . ., Vs which also span subspace S, but are mutually orthogonal (i.e., the dot product Vi . Vj = 0 for i /= j).  (Halmos, p. 127)

With homogeneous coordinates, ordinary hyperplanes g and h are orthogonal  if gN h = 0. Here gNh can be regarded as the dot product of the last n - 1 components of g and h, since the first component of gN is zero. The independent ordinary hyperplane matrix h = [h1, h2, . . ., hs] represents an ordinary flat S by intersections, and each of the hi has a normal hiN. By applying the Gram-Schmidt orthogonalization process to the hi they will be modified so hiNhj = 0,  i /= j.  Thus, geometrically, we have constructed s mutually orthogonal hyperplanes that intersect in S.

If we also normalize the hyperplanes hi then  hiN hi = 1 for all i. Then it follows that  hN h = Is. This is required for some of our matrix formulas below. In the procedure it is convenient to retain the same name for the original matrix and its orthonormalized output form.

Procedure C: Orthonormalize the columns of an ordinary independent hyperplane matrix g = (g1, g2, . . ., gs) so gN g = Is,  while preserving the null space and orientation of g.

Step 1.  For i = 1 to s do steps 2 and 3
Step 2.  If i > 1,  for j = 1 to i - 1:
Let d = gjN gi
Assign gi = gi - dgj
Step 3.  Normalize gi

III. TRANSFORMATION MATRICES

The following numbered formulas (M1, . . ., M16) give homogeneous transformation matrices T that effect familiar geometric transformations in a space of any dimension. If P is the homogeneous coordinates of a point, its transform P' is found by P' = PT.

We presume the transformations are originally designated by one or more of the following: (1) flats that are point-wise invariant under the transformation (e.g. an axis of a reflection), (2) scalar parameters (e.g. a dilation factor or angle), (3) one or more points and their known or desired transforms (e.g. for translation, the origin and its transform). Flats are designated by independent points (arranged as a point matrix) that generate the flat, or by independent hyperplanes (arranged as a hyperplane matrix) that intersect in the flat. If a formula requires a hyperplane matrix, Procedure B can be used to convert a point matrix representation of a flat to a hyperplane representation. If a formula requires an orthonormalized hyperplane matrix, Procedure C can be used to convert an ordinary hyperplane matrix to this form.

A. PROJECTION in Rn with null space the proper flat S1 and complementary flat S2 pointwise invariant.

Represent flat S1 by the independent point matrix C, and flat S2 (the "axis") by independent hyperplane matrix h. Then:

M1.       T = I - h(Ch)-1C (VanArsdale)

The projection of points in C is not defined. For ANY matrices C and h that can be multiplied: rank[Ch] = rank[C] - rank[range(C)^null(h)] (Methods). Thus the square matrix (Ch) in M1 is nonsingular since S1 and S2 are disjoint. For a discussion of projection and a derivation of M1 see Methods.  A "projection" is just one type of "projective transformation" - the latter including transformations such as translation that are unrelated to projection.

When h is a hyperplane and C is a single point not on h, M1 gives

M2.       T = I - hC/Ch  (VanArsdale)

More complicated coordinate expressions for projection appear in Hodge & Pedoe (p. 309) and other sources.

For an alternative representation of a projection from center S1 to the complementary axis S2 represent S1 by the r x n point matrix C and S2 by the (n - r) x n point matrix A.  Then

M3.       T = [C; A]-1 [0; A]       (Stolfi, p. 101)

Here "0" is the r x n matrix of all zeroes.

B. The GENERAL COLLINEATION mapping the n + 1 points P1, . . ., Pn+1 , no n of them dependent, to the n + 1 points Q1, . . . , Qn+1, no n of them dependent.
Let P = [P1; . . . ; Pn] and Q = [Q1; . . . ;Qn],  and set ci = qi / pi where pi is the ith coordinate of  Pn+1 P-1 and qi is the ith coordinate of Qn+1 Q-1.  Then

M4.       T = P-1 [c1Q1; . . .; cn Qn]

An equivalent matrix appears in Semple & Kneebone (p. 399) and other sources.

C. The AFFINE collineation mapping ordinary independent points P1, . . ., Pn to ordinary independent points Q1, . . . , Qn.
Normalize the points Pi and Qi, and let P = [P1; . . . ; Pn] and Q = [Q1; . . . ;Qn]. Then

M5.       T = P-1Q

Affine transformations map ideal points to ideal points. Some of the pairs of points (Pj, Qj) may be ideal if their representations are chosen correctly (Methods). This matrix, restricted to ordinary points and in an oriented context, appears in Stolfi (p. 158). Compare M5 to Snapper & Troyer (p. 97), where a nonhomogeneous approach requires, for n = 4, solving a system of 9 linear equations in 9 unknowns. M5 provides a matrix for other affine transformations presented below, though the resulting homogeneous transformation matrix is generally not as useful as those developed by the "axis-center" method.

D. The ISOMETRY mapping ordinary points P1, . . . Pm,  m = n -1, to congruent (superposable) points Q1, . . . , Qm.
Normalize the points Pi and Qi and let P = [P1; . . . ; Pm] and Q = [Q1 ; . . .; Qm]. Calculate f = Ph and g = Qh, f and g oriented, using procedure B. Normalize f and g. Then

M6.       T = [P; fN]-1 [Q; gN]       (VanArsdale)

With m = n-1 as above there are two isometries that effect the mapping, one direct and one indirect. Matrix M6 gives the direct isometry. For this case compare M6 to the complications of a nonhomogeneous method for three dimensions (only) in Laub & Shiflett. The above method can be easily adapted for m < n-1, there now being more than one normal to P and Q. For m = n use the method above for affinities, T = P-1Q.

E. The TRANSLATION mapping ordinary point P to ordinary point Q.
Let P and Q be normalized and w = [1; 0; . . .; 0] be the hyperplane at infinity.

M7.       T = I + w(Q - P).

This matrix is obvious in coordinate form (Roberts).

F. DILATION by factor d about ordinary flat S (of any rank < n).
Say S is represented by the point matrix P. Calculate g = Ph, a hyperplane representation of S, using Procedure B. Orthonormalize g using Procedure C. Then

M8.       T = I + (d-1) ggN (VanArsdale)

This is the transformation that leaves points P on S invariant and maps points P + V to P + dV, where V is any representation of a point on gN, the normal flat to S. The familiar examples of dilation are central dilation (S a point), and reflection in a plane (n = 4, S a plane, d = -1).  The following matrix for dilation does not require orthonormalization of g and depends on the fact that T is an affinity.

M9.       T = [P; gN] -1 [P; dgN]       (VanArsdale)

CENTRAL DILATION by factor d about the single ordinary point C (normalized) also has the representation

M10.       T = dI + (1-d)wC.

Example IV-D below shows how M10 can be used to analyze the composition of two central dilations.  This matrix is obvious in coordinate form.

For REFLECTION in ordinary flat S (of any rank < n) use d = -1 in M8 to get

M11.       T = I - 2ggN.

M11, at least for a hyperplane, has appeared in several sources.

G. The STRAIN or SHEAR which leaves ordinary hyperplane h point-wise invariant and maps point P to point Q, P and Q ordinary and distinct.
Normalize P and Q, then,

M12.       T = I + h(Q-P)/Ph (VanArsdale)

Since P and Q are distinct neither is on h, so Ph /= 0 and Qh /= 0. Strain and shear are affine, and determined by h, P and Q. For a shear, the line through P and Q must be parallel to h.

The SHEAR which leaves ordinary hyperplane h point-wise invariant and maps ideal point U to ideal point V.

M13.       T = I + h(V/Vh - U/Uh) (VanArsdale)

Since strain and shear are both affine, homogeneous transformation matrices based on M5 can be written for the above two transformations.

H. ROTATION about ordinary flat (axis) S of rank n - 2 by angle b.
Say S is represented by the point matrix P = [P1; . . . ;Pn-2 ]. Calculate an n x 2 hyperplane representation of S, g = Ph, g oriented, using procedure B. Then orthonormalize g using procedure C so gNg = I2 = [1, 0; 0, 1]. Calculate R = [cos b, sin b; -sin b, cos b]. Then

M14.       T = I + g(R - I2)gN       (VanArsdale)

A four dimensional example of the use of M14 appears below (IV-C).

Since rotation is affine we also have:

M15.       T = [P; gN]-1 [P; RgN]       (VanArsdale)

Some authors define a rotation on Rn as a direct isometry with an invariant point. When n > 4 such an isometry may require a composition of rotations as defined above. For n = 3 (the Euclidean plane) and n = 4, every rotation can be constructed as a composition of two reflections. For any dimension, such a rotation can be characterized as a mapping of hyperplanes.

The ROTATION that maps oriented hyperplane g to oriented hyperplane h, g and h ordinary and not parallel.

Normalize g and h. Let f = g + h (add components) and normalize f. Then,

M16.       T = [I - 2ggN] [I - 2ff`N]       (VanArsdale)

By an "oriented" hyperplane g we mean that the homogeneous coordinates of g are chosen so that if G is any ordinary point on g, G + gN lies in the "positive" side of g. T above maps the positive side of g to the positive side of h. The addition of the normalized hyperplane coordinates, f = g + h, gives a hyperplane f that bisects one of the two dihedral angles between g and h.

IV  LONGER EXAMPLES

AAN ORIENTED HYPERPLANE REPRESENTATION

Consider two points in two dimensional space (rank n = 3) with homogeneous coordinates P1 = [1,2,0] and P2 = [2,0,1]. These correspond to Cartesian coordinates (2,0) and (0,1/2) respectively. The point matrix P = [P1; P2], a 2 x 3 matrix, represents range(P), the line through P1 and P2. To find an oriented hyperplane (line) representation of range(P) use procedure B as follows.

(1) Form the 5 x 3 matrix [P; I] where I is the 3 x 3 identity matrix. Set the variable sgn = +1.

(2) Use elementary column operations on [P; I] to reduce P to an upper triangular matrix Q with ones on the diagonal, thereby changing [P; I] to [Q: E]. This can be done by (i) interchanging the second and third columns, (ii) subtracting twice the first column from the third, and (iii) adding four times the second column to the third. The interchange operation (i) requires we reverse the sign of sgn to -1.

(3) These operations give [Q; E] with Q= [1,0,0;  2,1,0] and E = [1,0,-2; 0,0,1; 0,1,4]. Now the third column of E (under the zero column of Q) is a 3 x 1 matrix h = [-2; 1; 4] that represents a hyperplane (line) containing P1 and P2, that is, P1h = P2h = 0. But to complete the third step in the procedure we must multiply h by sgn = -1 giving h = [2; -1; -4]. Then hN = [0, -1, -4] and the points P1, P2 and hN have positive orientation, i.e. det [P1; P2; hN] = 17 > 0. Such orientation is used to get the sense correct in formulas M6 and M14 - M16.

B. PROJECTION FROM A POINT TO A LINE

In the above example we found the hyperplane representation h = [2; -1; -4] (a 3 x 1 matrix) for the line through points P1 = [1,2,0] and P2 = [2,0,1]. We now seek a homogeneous matrix that will effect projection from the point C = [1, 1, 1] onto this line. This is given by formula M2: T = I - hC/Ch. Here Ch = -3 and hC = [2, 2, 2; -1, -1, -1; -4, -4, -4], a 3 x 3 matrix given row by row. Then T = 1/3 * [5,2,2; -1,2,-1; -4,-4,-1]. We can drop the initial factor of 1/3 since any nonzero multiple of matrix T does not change the projective transformation it represents. To find the projection of, for example, the origin O = [1, 0, 0] calculate OT = [5,2,2], which corresponds to Cartesian coordinates (2/5, 2/5).

C. ROTATION IN FOUR DIMENSIONS

We find a homogeneous matrix T that will effect a rotation in four dimensional space (n = 5), the point-wise invariant axis then being of rank n - 2 = 3, a plane. For this example we take the axis that contains the unit points on the x, y and z axes - a plane that does not pass through the origin. These three points have homogeneous coordinates P1 = [1,1,0,0,0], P2 = [1,0,1,0,0], and P3 = [1,0,0,1,0], and we take them in that order to fix a sense for measuring angles. Let P = [P1; P2; P3]. For the angle of rotation take b = 90 degrees.

First find an oriented hyperplane representation of the axis, g = Ph, using procedure B, to get g = [g1, g2], where g1 = [-1;1;1;1;0] and g2 = [0;0;0;0;1], both 5 x 1 column matrices. By the orientation feature of procedure B, det [P; gN] > 0. Orthonormalizing g (keeping the same variable name) using procedure C gives g = [g1, g2], where now g1 = [-r; r; r; r; 0], r = 3-1/2, g2 = [0;0;0;0;1] as before. Calculating: R = [cos b, sin b; -sin b, cos b] = [0, 1; -1, 0]. Then substituting in T = I + g(R - I2)gN (matrix M14) gives:

T = 1/3 * [3,1,1,1,-s;  0,2,-1,-1,s;  0,-1,2,-1,s;  0,-1,-1,2,s;  0,-s,-s,-s,0]

where the successive groups of five components are the rows of T and s = 31/2. The origin, O = (1,0,0,0,0), is rotated to OT = (3,1,1,1,-s).

D. THE COMPOSITION OF TWO CENTRAL DILATIONS

A projective transformation T may have a hyperplane h as an axis. Then we know it also has a center C of rank n - (n-1) = 1 (Methods). It is important for the classification of T whether this center point C lies on h or not. For example, say the axis h is the hyperplane at infinity, w. If C is on w then T is a translation; if C is not on w then T is a central dilation. The matrix representation of a central dilation about ordinary center point C by dilation factor d is T = dI + (1-d)wC; with scalar d /= 0 and d /= 1 and C normalized (M10). For two dilations their composition leaves w point-wise invariant and hence must be another central dilation or a translation. We can analyze this composition simply by multiplying homogeneous matrix representations.

Let      T1 = d1I + (1 - d1)wC1   (dilation about C1 by factor d1 )

and        T2 = d2I + (1 - d2)wC2   (dilation about C2  by factor d2 ).

Then     T1 *  T2  =  [d1I + (1 - d1)wC1] * [ d2I + (1 - d2)wC2]

=  d1d2I + d1(1-d2)wC2 + d2(1-d1)wC1 +  (1-d1)(1-d2)wC1wC2 .

The points C1 and C2 are both ordinary and normalized and so their first (homogeneous) coordinates are equal to 1. Thus in the last term above, the product C1w = 1. This gives:

(A)            T1 *  T2  =  d1d2I + d2(1-d1)wC1 + (1-d2)wC2

If d1d2 /= 1 then (A) can be written in the form

T1 *  T=  d1d2I   +  (1-d1d2) C

where C  =   [d2(1- d1) / (1-d1d2)] C1  + [(1- d2) / (1-d1d2)] C2.  This represents the central dilation with center C and dilation factor d1d2,  provided C is normalized. But this is the case since the homogeneous coordinate of C is  d2(1- d1) / (1-d1d2)  +  (1- d2) / (1-d1d2)  =  1.

If  d1d2 = 1 then (A) can be written in the form

T1 *  T=  I   +  w ( C' - C1)

where C'  =  d2C1 + (1 - d2 )C2.  This represents the translation that maps point C1 to point C' (M7).  Note C' is normalized since d2 + (1 - d2 ) = 1.  The center of this translation is the ideal point C' - C1= (1 - d2) [C2 - C1].  This could have been anticipated since it is clear the composition of the dilations leaves the line through C1 and C2 invariant, and the invariant center is the intersection of this line with the invariant hyperplane at infinity, w.

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REFERENCES

Ayres, F. Jr., Matrices, Schaum's Outline Series, New York, 1962.

Coxeter, H.S.M., The Real Projective Plane (2nd ed.), Cambridge, 1961.

Fishback, W.T., Projective and Euclidean Geometry (2nd ed.), John Wiley & Sons, New York, 1969.

Halmos, P.R., Finite-Dimensional Vector Spaces, (2nd ed.), Van Nostrand, New York, 1958.

Hodge, W.V.D & Pedoe, D.,  Methods of Algebraic Geometry (Vol. 1), Cambridge Univ. Press, 1968.

Laub, A.J. & Shiflett, G.R., A linear algebra approach to the analysis of rigid body displacement from initial and final position data. J. Appl. Mech. 49, 213-216, 1982.

Pedoe, D., Geometry, Dover, New York, 1988.

Roberts, L.G., Homogeneous Matrix Representation and Manipulation of N-dimensional Constructs. MIT Lincoln Laboratory, MS 1405, May 1965.

Semple, J.G. & Kneebone, G.T.,  Algebraic Projective Geometry, Clarendon Press, Oxford, 1952.

Shilov, G.E., Linear Algebra, Dover Publications, New York, 1977.

Snapper, E. & Troyer, R.J.,  Metric Affine Geometry. Academic Press, 1971.

Stolfi, J., Oriented Projective Geometry, Academic Press, 1991.

VanArsdale, D., Homogeneous Transformation Matrices for Computer Graphics, Computers & Graphics, vol. 18, no. 2, 177-191, 1994.

Homogeneous Coordinates: Methods
Theorems and methods utilizing homogeneous coordinates - many unpublished. Companion to this site.

Transformation of Coordinates
Uses coordinates to prove some classical theorems in plane projective geometry.

Britannica.com
Some history of projective geometry, both synthetic and analytic methods, basics of  homogeneous coordinates.

Math Forum - Projective geometry
Internal links to articles on projective geometry at various levels. Useful online resource.

Geometric transformations
Elementary 2D and 3D transformations, including affine, shear, and rotation.

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