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
^{-1}h - 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**

*R ^{n} = *the

n = the

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

S_{1 }__c__ S_{2} means set S_{1} is
contained in, or equal to, set S_{2}.

P = [X_{1}, X_{2}, ..., X_{n}]
= the homogeneous coordinates of **point** P, a 1x n row
matrix, P / = 0.

For any nonzero constant c, both P and cP = [cX_{1},
cX_{2}, . . ., 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.

X_{1 }is the **homogeneous coordinate**
of point P. Often in the literature the homogeneous coordinate
of a point is placed last (X_{n}) instead.

If X_{1 }/= 0, P is **ordinary**
and corresponds to the Cartesian point (X_{2}/X_{1},
. . ., X_{n}/X_{1}).

P (ordinary) is **normalized **if X_{1 }= 1.

If X_{1 }= 0, P is **ideal**
and may represent the direction of the Cartesian vector (X_{2},
. . ., X_{n}). In a projective context P is the same
ideal point as -P = (0, -X_{2}, . . ., -X_{n}),
but in practice we often distinguish these directions.

P (ideal) is normalized if (X_{2})^{2} + .
. . + (X_{n})^{2} = 1.

Example: In *R ^{3}* 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

h = [Y_{1};Y_{2}; . . .; 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 = [cY_{1};
cY_{2}; . . .; 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 = Y_{2}^{2}+.
. . + Y_{n}^{2}:

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 *R ^{4}* (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.

S = a **flat of rank r** = the set of all
points of the form c_{1}P_{1} + . . .+ c_{r}P_{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.
*R ^{n}* is a flat of rank n. These two flats are improper, all others are proper.

Flat S is

S_{1 }^ S_{2} = the intersection
(meet) of flats S_{1} and S_{2 }, also a flat.

S_{1}v S_{2} = the union (join) of flats S_{1}
and S_{2 }, also a flat.

rank(S_{1}) + rank(S_{2}) = rank(S_{1}v
S_{2}) + rank(S_{1 }^ S_{2})
(Ayres, p. 87)

In *R ^{n}* flats S

P = a **point matrix** = any r x n matrix, P
= [P_{1}; P_{2}; . . .; P_{r}], r >=
1. The 1 x n matrices P_{i} are the rows of P, and if
nonzero represent points in *R ^{n}*. Point
matrices are represented by upper case letters, or by
upper case superscripts.

P **represents **the flat spanned by its
rows P_{1}, P_{2}, . . ., P_{r}.
This flat is also called the **row space** of P, or
the **union** of P_{1}, P_{2}, . . .,
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 P_{1}, P_{2},
. . . , P_{r} are independent.

Say P_{1}, . . ., P_{n} are n
ordered independent points in *R ^{n}* and let P =
[P

h = a **hyperplane matrix** = any n x s
matrix, h = [h_{1}, h_{2}, . . . , 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 h_{1}, . . ., 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 h_{1}, h_{2},
. . ., h_{s} are independent.

If the rank of independent hyperplane matrix h in R^{n} is s, the rank of the flat it represents by
intersections is n - s.

Example: In *R ^{4}* 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 = [h

A projective
transformation f on *R ^{n}* (real
projective space of dimension n-1) is a mapping of a set of
points of R

(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

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 collineation T, hyperplane h is mapped to hyperplane T

Collineations map straight lines to straight lines.

For a projective transformation f on *R ^{n}*
represented by matrix T:

The

Range(T) is a flat with rank equal to rank(T).

The

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

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 = [h_{1}, h_{2}], we
define g^{N} = [ h_{1}^{N}; h_{2}^{N
}], an ideal point matrix with two rows. When g is
"orthonormalized" (see Procedure
C) then g^{N }g = I_{2}. 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 *R ^{n}* 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 = P

An additional condition may be imposed on g. If
the axis of a rotation is designated as the line through points
P_{1} and P_{2} this implies a sense of rotation
opposite that if the two points are taken in the order P_{2 }and
P_{1}. In
converting this line to a hyperplane representation h = [h_{1},
h_{2}] we need to assure that the orientation of the
four points P_{1}, P_{2},_{ }h_{1}^{N},
and h_{2}^{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 v_{1},
v_{2}, . . ., 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 V_{1}, V_{2}, . . ., 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^{N}h 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 = [h_{1},
h_{2}, . . ., 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}^{N}h_{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 = (g_{1}, g_{2}, . . ., 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}

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 *R ^{n }*with null space the proper flat S

Represent flat S_{1} by the independent
point matrix C, and flat S_{2} (the "axis") by
independent hyperplane matrix h. Then:

**M1. T = I
- h(Ch) ^{-1}C **(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 S_{1} and
S_{2 }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 S_{1} to the complementary axis S_{2 }represent
S_{1} by the r x n point matrix C and S_{2} 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 P_{1}, . .
., P_{n+1 }, no n of them dependent, to the n + 1
points Q_{1}, . . . , Q_{n+1}, no n of them
dependent.

Let P = [P_{1}; . . . ; P_{n}] and Q = [Q_{1};
. . . ;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}
[c_{1}Q_{1}**;

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

C. The **AFFINE **collineation
mapping ordinary independent points P_{1}, . . ., P_{n}
to ordinary independent points Q_{1}, . . . , Q_{n}.

Normalize the points P_{i} and Q_{i}, and let P
= [P_{1}; . . . ; P_{n}] and Q = [Q_{1};
. . . ;Q_{n}]. Then

**M5. T = P ^{-1}Q**

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 P_{1}, . . . P_{m}, m = n
-1, to congruent (superposable) points Q_{1}, . . . , Q_{m}.

Normalize the points P_{i} and Q_{i} and let P =
[P_{1}; . . . ; P_{m}] and Q = [Q_{1 };
. . .; 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^{-1}Q.

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.

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

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 point matrix P = [P_{1}; . .
. ;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^{N}g
= I_{2} = [1, 0; 0, 1]. Calculate R = [cos b, sin b;
-sin b, cos b]. Then

**M14.
T = I + g(R - I _{2})g^{N}**
(VanArsdale)

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

Since rotation is affine we also have:

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

Some authors define a rotation on *R ^{n}*
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 - 2gg ^{N}] [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 + g^{N} 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.

**A**. **AN
ORIENTED HYPERPLANE REPRESENTATION**

Consider two points in two dimensional space
(rank n = 3) with homogeneous coordinates P_{1 }=
[1,2,0] and P_{2 }= [2,0,1]. These correspond to
Cartesian coordinates (2,0) and (0,1/2) respectively. The point
matrix P = [P_{1}; P_{2}], a 2 x 3 matrix,
represents range(P), the line through P_{1 }and P_{2}.
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 P_{1
}and P_{2}, that is, P_{1}h = P_{2}h
= 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 P_{1}, P_{2 }and h^{N}
have positive orientation, i.e. det [P_{1}; P_{2};
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 P_{1 }= [1,2,0] and P_{2 }=
[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 P_{1} =
[1,1,0,0,0], P_{2} = [1,0,1,0,0], and P_{3} =
[1,0,0,1,0], and we take them in that order to fix a sense for
measuring angles. Let P = [P_{1}; P_{2}; P_{3}].
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 = [g_{1},
g_{2}], where g_{1 }= [-1;1;1;1;0] and g_{2}
= [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 = [g_{1}, g_{2}], where now
g_{1} = [-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 - I_{2})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 T_{1} =
d_{1}I + (1 - d_{1})wC_{1}
(dilation about C_{1} by factor d_{1} )

and_{
}T_{2} = d_{2}I + (1 - d_{2})wC_{2
}(dilation about C_{2} by factor d_{2}
).

Then T_{1}
* T_{2} = [d_{1}I + (1 - d_{1})wC_{1}]
* [ d_{2}I + (1 - d_{2})wC_{2}]

= d_{1}d_{2}I + d_{1}(1-d_{2})wC_{2}
+ d_{2}(1-d_{1})wC_{1} + (1-d_{1})(1-d_{2})wC_{1}wC_{2}
.

The points C_{1} and C_{2} are
both ordinary and normalized and so their first (homogeneous)
coordinates are equal to 1. Thus in the last term above, the
product C_{1}w = 1. This gives:

**(A) **
T_{1} * T_{2} = d_{1}d_{2}I
+ d_{2}(1-d_{1})wC_{1} + (1-d_{2})wC_{2}

If d_{1}d_{2} /= 1 then (A) can
be written in the form

T_{1} * T_{2 }= d_{1}d_{2}I
+ (1-d_{1}d_{2}) C

where C = [d_{2}(1- d_{1})
/ (1-d_{1}d_{2})] C_{1} + [(1- d_{2})
/ (1-d_{1}d_{2})] C_{2}. This
represents the central dilation with center C and dilation
factor d_{1}d_{2},_{ }provided C
is normalized. But this is the case since the homogeneous
coordinate of C is d_{2}(1- d_{1}) / (1-d_{1}d_{2})
+ (1- d_{2}) / (1-d_{1}d_{2})
= 1.

If d_{1}d_{2} = 1 then (A)
can be written in the form

T_{1} * T_{2 }= I
+ w ( C' - C_{1})

where C' = d_{2}C_{1}
+ (1 - d_{2} )C_{2}. This represents the
translation that maps point C_{1} to point C'
(M7). Note C' is normalized since d_{2} + (1 - d_{2}
) = 1. The center of this translation is the ideal point
C' - C_{1}= (1 - d_{2}) [C_{2} - C_{1}].
This could have been anticipated since it is clear the
composition of the dilations leaves the line through C_{1}
and C_{2} 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.

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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