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.

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

IV.   Longer Examples
A. An oriented hyperplane representation  B. Projection from a point to a line  C. Rotation in four dimensions  D. Composition of central dilations

V.  References  and Links

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.   I_s = 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 S₁is contained in, or equal to, set S₂.

B. POINTS

P = [X₁, X₂, . . ., X_n] = the homogeneous coordinates of point P, a 1 x n row matrix, P / = 0.
For any nonzero constant c, both P and cP =  [cX₁, cX₂, . . ., cX_n] 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 (X_n) instead.

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

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

For h as above, let C = Y₂²+. . . + Y_n²:
     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 + . . .+ c_rP_r, the c_i any constants (not all zero), the P_i some r <= n fixed independent points.  As in linear algebra, the P_i  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; . . .; P_r], r >= 1. The 1 x n matrices P_i 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, . . ., P_r. This flat is also called the row space of  P, or the union of  P₁, P₂, . . ., P_r.  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, . . . , P_r are independent.

Say P₁, . . ., P_n are n ordered independent points in Rⁿ and let P = [P₁; . . .; P_n]. Then P₁, . . ., P_n 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, . . . , h_s], s <= n. The n x 1 matrices h_i 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, . . ., h_s and may be designated by null(h) or h^P.

h is ordinary if h^P is ordinary.
h is independent if its columns h₁, h₂, . . ., h_s 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.

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  T^P.
    For any T,  rank[T] + rank[null(T)] = n

II. THREE PROCEDURES

A. The NORMAL, h^N, 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 h^N.  For a hyperplane matrix g with more than one column, say g = [h₁, h₂], we define g^N = [ h₁^N; h₂^N ], an ideal point matrix with two rows. When g is "orthonormalized" (see Procedure C) then g^N g = I₂. Normals are used in several of the transformation matrices below.

Procedure A: Find the normal, h^N, 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 h^N 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, P^h.

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 would 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 = P^h. 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 of the line through 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,  h₁^N and h₂^N, 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 = P^h such that:  (1) g represents S by intersections, and (2) if P is ordinary, det [P; g^N] > 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 = P^h 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, . . ., v_s span an s-dimensional subspace, S, by linear combinations. The familiar Gram-Schmidt orthogonalization process uses linear combinations of the v_i to produce vectors V₁, V₂, . . ., V_s which also span subspace S, but are mutually orthogonal (i.e., the dot product V_i . V_j = 0 for i /= j).  (Halmos, p. 127)

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

If we also normalize the hyperplanes h_i then  h_i^N h_i = 1 for all i. Then it follows that  h^N h = I_s. 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, . . ., g_s) so g^N g = I_s,  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 = g_j^N g_i
            Assign g_i = g_i - dg_j
Step 3.  Normalize g_i

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, . . ., P_n+1 , no n of them dependent, to the n + 1 points Q₁, . . . , Q_n+1, no n of them dependent.
Let P = [P₁; . . . ; P_n] and Q = [Q₁; . . . ;Q_n],  and set c_i = q_i / p_i where p_i is the i^th coordinate of  P_n+1 P^-1 and q_i is the i^th coordinate of Q_n+1 Q^-1.  Then

M4.       T = P^-1 [c₁Q₁; . . .; c_n Q_n]

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

C. The AFFINE collineation mapping ordinary independent points P₁, . . ., P_n to ordinary independent points Q₁, . . . , Q_n.
Normalize the points P_i and Q_i, and let P = [P₁; . . . ; P_n] and Q = [Q₁; . . . ;Q_n]. Then

M5.       T = P^-1Q

Affine transformations map ideal points to ideal points. Some of the pairs of points (P_j, Q_j) 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, . . . P_m,  m = n -1, to congruent (superposable) points Q₁, . . . , Q_m.
Normalize the points P_i and Q_i and let P = [P₁; . . . ; P_m] and Q = [Q₁ ; . . .; Q_m]. Calculate f = P^h and g = Q^h, f and g oriented, using procedure B. Normalize f and g. Then

M6.       T = [P; f^N]^-1 [Q; g^N]       (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 = P^h, a hyperplane representation of S, using Procedure B. Orthonormalize g using Procedure C. Then

M8.       T = I + (d-1) gg^N (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 g^N, 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; g^N] ^-1 [P; dg^N]       (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 - 2gg^N.

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 ordered point matrix P = [P₁; . . . ;P_n-2 ]. Calculate an n x 2 hyperplane representation of S, g = P^h, g oriented, using procedure B. Then orthonormalize g using procedure C so g^Ng = I₂ = [1, 0; 0, 1]. Calculate R = [cos b, sin b; -sin b, cos b]. Then

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

Or since rotation is affine:

M15.       T = [P; g^N]^-1 [P; Rg^N]       (VanArsdale)

Rotation requires n > 2. For n > 4 some authors define a rotation more generally as a direct isometry with an invariant point.  These result from the composition of rotations as defined above. A four dimensional example of the use of M14 appears below (IV-C).

The ROTATION that maps oriented hyperplane g to oriented hyperplane h, g and h ordinary.
Normalize g and h. Let f = g + h (add components) and normalize f. Then,

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

By an "oriented" hyperplane g we mean that homogeneous coordinates of g are chosen to determine a positive side (half space) of g as follows: if G is any ordinary normalized point on g, then G + g^N lies on the positive side of g. T above maps the positive side of g to the positive side of h.  Matrix M16 uses the well known fact that reflection in two hyperplanes is a rotation through twice their angle of incidence [Pedoe, p. 63]. The addition of the normalized hyperplane coordinates, f = g + h, gives a bisecting hyperplane f between g and h.

IV  LONGER EXAMPLES

A. AN 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 h^N = [0, -1, -4] and the points P₁, P₂ and h^N have positive orientation, i.e. det [P₁; P₂; h^N] = 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 = P^h, 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; g^N] > 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 , and 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)g^N (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 = 3^1/2. 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 *  T₂  =  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 *  T₂  =  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.

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.

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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