This is a selection of basic facts, theorems and methods that utilize homogeneous coordinates. Some of these results have been previously published but are rarely seen in textbooks. Others come from my article "Homogeneous Transformation Matrices for Computer Graphics," Computers & Graphics, vol. 18, no. 2, 1994.  The presentation is at an undergraduate level, but the reader should have had some exposure to analytic projective geometry and a course in linear algebra. However beware, many facts change when you go from vector spaces and Cartesian coordinates to homogeneous coordinates. For example, "flats," unlike subspaces, need not contain the origin, and a point is a flat. And an eigenvector becomes an invariant point, and its eigenvalue is not fixed.

This is a companion site to Homogeneous Transformation Matrices (HTM), a cookbook presentation of matrices that effect familiar geometric transformations. Many of the conventions and notation used here are introduced in HTM, so you should link there first and read sections I and II. Included here are methods sufficient to derive all the matrix formulas in HTM (referred to as M1, M2, . . ., M16). The references and links presented at the end here are identical to the ones in HTM. 

TOPICS

1.    Some Basic Facts
2.    Duality
3.    Oriented Content
4.    Rank of the Product of Two Matrices
5.    Affine Collineations
6.    Null Space
7.    Intersections
8.    Invariant Flats
9.    Axis-Center Form
10.  Singular Transformations
11.  Three Dependent Points
12.  A Perspective Matrix
13.  References and Links

1.   Some Basic Facts

F1.  If points P₁, . . ., P_n are independent in Rⁿ, any point X has a representation of the form X = c₁P₁ + . . . + c_n P_n where the c_i are not all zero. With P = [P₁; . . .; P_n] (an nxn matrix, see HTM) and C =  [c₁, . . .,c_n] this is X = CP, so C = XP^-1. 

F2.   If the n x s (s <= n) matrix h has independent columns there exists an s x n matrix H, the left inverse of h,  such that Hh = I_s,  I_s the s x s identity matrix. Similarly the r x n (r <=n) independent matrix P has a right inverse.  [Shilov, p. 98]

F3.   Let P = [P₁; . . .; P_r] be an  independent r x n point matrix. (1) The single point Q is contained in range(P) if and only if there exists constants k₁, . . ., k_r such that Q = k₁P₁ + . . . + k_rP_r, which is Q = KP, with K the 1 x r matrix (k₁, . . ., k_r). (2) For an r' x n point matrix Q, range(Q) c  range(P) if and only if Q = KP for some r' x r matrix K.

F5.   In Rⁿ an ideal point U is normalized if UU^t = 1. If ideal points U and V are normalized then UV^t = cos x where x is the angle between the two lines connecting any ordinary point to U and to V.  The "angle between U and V" is defined as x.  If UV^t = 0 then U and V are orthogonal.

2.   Duality

Duality in plane projective geometry is described as follows by Coxeter:

"The principle of duality (in two dimensions) asserts that every definition remains significant, and every theorem remains true, when we interchange point and line, and join and intersection. To establish this principle we merely have to observe that the axioms imply their own duals."  (Coxeter, p. 15)

Example:
Say P and Q are distinct points, not both on hyperplane h. Then the point X = (Qh)P - (Ph)Q is the intersection of the line through P and Q with h.
This follows since by its form X is on the line through points P and Q, and Xh = 0. The dual is:
Say p and q are distinct hyperplanes, not both containing point H. Then the hyperplane x = (qH)p - (pH)q is the join of the intersection of p and q with H.

Facts F3 and F3D are dual, as are Theorems 5 and 5D.

3.   Oriented Content

A student's first encounter with homogeneous coordinates is often the following:

Theorem. In the Cartesian plane, distinct points P₁= (x₁, y₁), P₂= (x₂, y₂), P₃= (x₃, y₃) are collinear if and only if det [1, x₁, y₁;  1, x₂, y₂; 1, x₃, y₃] = 0.

In the 3 x 3 matrix above the rows (1, x₁, y₁), (1, x₂, y₂), (1, x₃, y₃) are the normalized homogeneous coordinates of points P₁, P₂, P₃. This interpretation is not developed in the texts I have examined, nor do linear algebra textbooks give the following:

Theorem 1: In Rⁿ the oriented content of the simplex formed by the n ordered independent points P₁, . . ., P_n, each ordinary and normalized, is  [1/(n-1)!] det [P₁; . . .; P_n].     [Stolfi, p.164]

Orientation here matches what we expect in one, two and three dimensions. For example, in the plane the determinant will be positive if P₁, P₂ and P₃ form a counterclockwise triangle. When the points are dependent (collinear in R³) the determinant is zero.

4.   The Rank of the Product of Two Matrices

In matrix theory the rank of a matrix A is the maximum number of independent rows (or columns) that can be selected. This is equal to the dimension of the subspace spanned by the rows of A. The following theorem appears in every basic textbook on matrices (e.g. Ayres, p. 43):

Theorem: Let A be an r x n matrix and B be an n x s matrix. Then
(1) rank (AB) <= rank (A)
(2) rank (AB) <= rank (B).

Interpreting the components of the matrices as homogeneous coordinates we have the following more specific theorem (VanArsdale).

Theorem 2: Let A be an r x n matrix and B be an n x s matrix. Then
(1)  rank (AB) = rank (A) - rank [range(A) ^ null(B)]
(2)  rank (AB) = rank (B) - rank [range(B^t) ^ null(A^t)].

To prove (1), using concepts from linear algebra, extend a  basis Y₁, . . ., Y_j of  range(A) ^ null(B) to range(A),  producing Y₁, . . ., Y_j, Y_j+1, . . ., Y_j+k. Then Y_j+1B, . . ., Y_j+kB is a basis for range(AB). Thus rank[range(A)] = rank(A) = rank[range(A) ^ null (B)] + rank(AB) as required. The second part of the theorem follows after transposing the first. Here B^t is the transpose of B.

Theorem 2 gives a geometric interpretation of  how much less rank (AB) is from rank (A), and we see at once that if the flats represented by A and B are disjoint the two ranks are equal.

5.   Affine Collineations

An affine collineation (affinity) maps ideal points to ideal points. Geometrically this means parallel lines are mapped to parallel lines. Most familiar linear transformations are affine, for example rotation, dilation, translation, reflection and shear. A nonsingular matrix T represents an affinity if and only if its first column equals [k; 0; . . .; 0], k /= 0. If k = 1 T is normalized and T maps ordinary normalized points to ordinary normalized points.

An affine collineation on Rⁿ is determined by its mapping of n independent ordinary points (Snapper & Troyer, p. 93). But it is useful to allow ideal points as follows.

Theorem 3:  Let P₁, . . ., P_n and Q₁, . . ., Q_n be two sets of points in Rⁿ such that: (1) for each i,  P_i and Q_i have the same homogeneous coordinate,  (2) the P_i and the Q_i are independent (thus at least one pair, say P_k and Q_k, are ordinary). Let P = [P₁; . . .; P_n] and Q = [Q₁; . . .; Q_n]. Then the matrix T = P^-1Q is a normalized representation of the unique affine collineation, f, for which: (a) (P_i)f  =  Q_i for all P_i and Q_i ordinary, and (b) (P_k + P_j)f  = Q_k + Q_j for all P_j and Q_j ideal.

Proof:  The mapping conditions on f specify a unique affinity since the P_i and P_k+ P_j constitute n independent ordinary points, likewise their images Q_i and Q_k + Q_j.  And matrix T effects these mappings since PT = Q (see F4). Finally we need to show that T is affine and normalized. Since the first (homogeneous) columns of P and Q are equal, Pw = Qw, so w = P^-1Qw = Tw which says the first column of T is w = (1: 0; . . . ;0).  (Stolfi, p.158, for all ordinary points).

The conditions in Theorem 3 mean that if we use ideal points P_j and P_jf = Q_j to specify an affine transformation we must choose representations of P_j and Q_j that work when added to some ordinary point. Usually the ordinary point P_k in the theorem will be an invariant point on the axis of  f. For a simple example, the reflection f about the x-axis in R³ maps (0,0,1)f = (0,0,1),  (1,0,1)f = (1,0,1) and  (0,1,0)f  = (0, -1, 0) = Q_j. Using these coordinates in Theorem 3 will give a correct matrix for f, T = [1,0,0; 0,-1,0; 0,0,1]. But if for (0,1,0)f we had used Q_j = (0,1,0), which is projectively equivalent to point (0,-1,0), an incorrect matrix would result.

Let the rank n - 2 ordinary flat S be represented by the point matrix P = [P₁; . . .; P_n-2] and by hyperplane matrix g = [g₁, g₂] = P^h, g oriented and orthonormalized. Then the rotation f by angle b about axis S has the representation  T = [P; g^N]^-1 [P; Rg^N]  where R = [cos b, sin b; -sin b, cos b]  (M15 in HTM). This can be verified by noting (see F4): [P; g^N]T = [P; Rg^N], thus P_iT = P_i for i = 1, . . ., n-2  and

     [g₁^N; g₂^N] T = g^NT = Rg^N  = [(cos b) g₁^N+ (sin b) g₂^N; (-sin b) g₁^N + (cos b) g₂^N]

Thus T has mapped n - 2 + 2 =  n independent points correctly, and since f is affine by Theorem 3 this insures T represents f.  If one lacks confidence about rotating the ideal points g₁^N_and g₂^N their coordinates can both be added to an ordinary invariant point on axis S to return to the familiar. Theorem 3 can also be used to derive equations M3, M5, M6, and M9 in HTM.

6.   Null Space

In HTM we introduced the notation P^h for the independent hyperplane matrix that represents the same flat (by intersections) that a given point matrix P represents (by unions). Procedure B in HTM provides a method to calculate P^h by elementary column operations. It is also useful to be able to reverse this process and construct an independent point matrix, Q, that represents the same flat that a given hyperplane matrix g represents. This is simply a point representation of the null space of g, Q = null(g), or Q = g^P. Procedure B can be used to calculate g^P as follows.
 

Procedure D: For a hyperplane matrix g, find an independent point matrix
Q = g^P such that range(Q) = null(g).

Step 1. Transpose g (getting g^T).
Step 2.  Use Procedure B to find (g^T)^h
Step 3.  Transpose  (g^T)^h  to find Q =  g^P
Step 4. (Optional). Normalize the rows of Q.

7.   Intersections

Homogeneous coordinates are at their best in finding the intersection of two flats. To perform the calculations involved it is necessary to make two simple modifications to the two procedures (B and D) that convert point representations to hyperplane representations (P^h), and visa versa (h^P).

Procedure B (and D) modifications:
(1) Allow the input matrix P (or h) to have dependent rows (or columns).
(2) Allow the input and output of "improper" matrices: (i) a point matrix with no rows representing the null flat, and (ii) a hyperplane matrix with no columns, representing all of Rⁿ.

Theorem 4.       Q₁ ^ Q₂  =  [Q₁^h, Q₂^h] ^P

This works in a space of any dimension, for flats of any dimension, for parallel flats, and for intersections of any dimension including the null flat for disjoint inputs. Coding requirements are simple: just reduction to a standard form by elementary column operations, the rank n variable. If the intersection is a single point that is all you will get, since as specified the procedure for h^P produces a matrix with independent rows. This obvious and extremely general method has probably been known long before 1968 (Hodge & Pedoe, p. 189), but it gets little or no attention in basic texts and packaged programs.

The capability to find intersections and unions makes the following two constructions in three dimensional Euclidean space (R⁴ ) almost a matter of definition.

In R⁴, given two skew lines l₁ and l₂  and a point X on neither line, find the line through X  that meets both l₁ and l₂.  Answer: [(l₁ v X) ^ l₂] v X = (l₁ v X) ^ (l₂ v X).

In R⁴, given two ordinary skew lines l₁ and l₂, find the line that meets both l₁ and l₂ at right angles.  Answer: Let X = [(l₁)^N ^ (l₂)^N]  in the above construction. This finds the shortest line segment connecting the two skew lines.

8.   Invariant flats

Say a projective transformation f on Rⁿ is represented by the homogeneous matrix T. A point P is invariant under f  if  PT = kP, k any nonzero constant. The scalar k is called the eigenvalue of P with respect to T. In contrast to nonhomogeneous matrices, if homogeneous matrix T is multiplied by a nonzero constant c it is unchanged as a transformation matrix, but the eigenvalues of all invariant points under T are multiplied by c. Thus T can be "scaled" so the eigenvalue of P is 1. If for a point P, PT = 0, rather than stating "f maps P to a vector of all zeros" it is more accurate to state "P is not in the domain of f." Or we may regard P as mapped by f to the null flat. For homogeneous matrices we do not regard 0 as an eigenvalue.

A flat, considered as a set of points S, is invariant under a mapping f if f is a bijection on S. If each point of S is itself invariant the flat is said to be point-wise invariant (P-invariant). All points on a P-invariant flat have the same eigenvalue, and conversely all invariant points with equal eigenvalues under a nonsingular projective transformation constitute a flat. An axis of f is a P-invariant proper flat that is not strictly contained in a P-invariant flat. I call the common eigenvalue of points on an axis the P-eigenvalue of the axis to distinguish this number from the eigenvalue of hyperplanes.

9.   Axis - Center Form

Theorem  5.  Say a projective transformation f on Rⁿ has an axis S of rank r and h is any independent n x s hyperplane matrix representation of S, s = n - r. Then f has a matrix representation of the form

                                              T = I + hC

where C is some independent point matrix. If f is a collineation, C represents a center of f.   (VanArsdale)

Proof:  Since the common eigenvalue of points on axis S is nonzero, any matrix representation T of f can be scaled so this P-eigenvalue is 1. Then any invariant point Q with eigenvalue 1 must be on h since h is an axis, and we have Q(T - I) = 0 if and only if Qh = 0. Interpreting T - I as a hyperplane matrix we get (T - I)^P = h^P and so by F3D,  T - I = hC, C some s x n point matrix. Now rank(C) > rank(hC) = rank(T- I) = rank(h) = s, and since C has s rows these must be independent.

It remains to show that if f is a collineation, range(C) is a center of  f. Every hyperplane g of Rⁿ is in the domain of a collineation f, and those that contain C are invariant under f with eigenvalue 1 since Tg = (I + hC)g = g using Cg = 0. And any invariant hyperplane g with eigenvalue 1 contains range(C), for Tg = [I + hC]g = g implies hCg = 0 which requires Cg = 0 since h has a left inverse. This shows there is no h-invariant proper subflat of range(C), hence C represents a center of  f.

Theorem 5D. Say a projective transformation f on Rⁿ has a center S of rank s and C is any independent s x n point matrix representation of S. Then f has a matrix representation of the form

                                              T = I + hC

where h is some independent hyperplane matrix. If f is a collineation, h represents an axis of f.

Proof. Since the common eigenvalue of hyperplanes containing center S is nonzero, any matrix representation T of f can be scaled so this h-eigenvalue is 1. Then any invariant hyperplane g with eigenvalue 1 must contain C since C is a center, and we have (T - I)g = 0 if and only if Cg = 0. Interpreting T - I as a point matrix we get range(T - I) = range(C) and so by F3, T - I = hC, h some n x s hyperplane matrix. Now rank(h) > rank(hC) = rank(T- I) = rank(C) = s, and since h has s columns these must be independent.

It remains to show that if f is a collineation, h^P is an axis of  f. Every point P of Rⁿ is in the domain of a collineation f, and those in h^P are invariant under f with eigenvalue 1 since PT = P(I + hC) = P using Ph = 0. And any invariant point P with eigenvalue 1 lies in h^P, for PT = P[I + hC] = P implies PhC = 0 which requires Ph  = 0 since C has a right inverse. This shows there is no P-invariant flat that properly contains h^P, hence h represents an axis of  f.

(9A)          T = I + hMC

and further conditions on f may allow us to solve for matrix M. When rank(h) = rank(C) = 1, the "matrix" M is a scalar. This "axis-center" method can be used to derive matrices  M1, M2, M7, M8, M10, M12, M13 and M14 in HTM. In the next section we illustrate its use in deriving a matrix for singular projection (M1).

10.   Singular Transformations 

If a projective transformation f on Rⁿ maps at least one point to null (i.e. there is at least one point not in the domain of f), then we call f singular, and it is represented by a singular n x n matrix T.

(10A)           T = I - h(Ch)^-1C

for a matrix representation of a general projection. When the axis h is a single hyperplane this reduces to T = I - hC/Ch, an attractive formula that I have not been able to find published prior to 1994 (VanArsdale). Please inform me if you know of a prior source for this. More complicated equivalent expressions do appear. Hodge and Pedoe give one using Grassmann coordinates (p. 309).

We have, in effect, defined a projection as a projective transformation with complementary null space and axis. Other equivalent definitions appear in the literature [Hodge and Pedoe, p. 283; Halmos, p. 73].

11.   Three dependent points.

In linear algebra vectors  V₁, . . ., V_r are dependent if constants c₁, . . ., c_r exist, not all zero, such that c₁V₁ + . . . + c_rV_r = 0. To actually find the constants c_i requires the solution of a system of  homogeneous linear equations, which is easy enough to do. But there is usually no convenient explicit expression for these constants, nor a geometrical interpretation of them. In projective space, Rⁿ, we have for homogeneous coordinates:

Theorem 6.     If three points P₁, P₂, P₃ are dependent and h₁, h₂ are any two hyperplanes then

                 (d₂₁d₃₂ - d₂₂d₃₁)P₁  + (d₃₁d₁₂ - d₃₂d₁₁)P₂ + (d₁₁d₂₂ - d₁₂d₂₁)P₃ = 0

where d_ij = P_ih_j.

To prove this we can write the lengthy dependence relation as the symbolic determinant

D = det [P₁, P₂, P₃;  P₁h₁, P₂h₁, P₃h₁;  P₁h₂,  P₂h₂, P₃h₂ ].

Here the first row, [P₁, P₂, P₃], consists of three points each with n homogeneous coordinates, so D can not be evaluated as a number.  But when we consider each of the  coordinates in turn D becomes a legitimate determinant. And each of these n determinants will be zero. For since the points are dependent, one is a linear combination of the others, say P₃ = c₁P₁ + c₂P₂. Substituting this for P₃ everywhere in D will produce a determinant with dependent columns for each of the n coordinates of the points.

One can write corollaries, duals and extensions of Theorem 7.

Corollary 6A.    If normalized ordinary points P₁, P₂, P₃ are dependent and h is any hyperplane then  (d₂-d₃)P₁ + (d₃-d₁)P₂ + (d₁-d₂)P₃ = 0  where d_i = P_ih.

This follows from Theorem 7 by letting h₁ = h and  h₂ = w, the hyperplane at infinity, for then  P_ih₂ = d_i2 = 1. Here the d_i have a geometrical interpretation for h ordinary and normalized:  d_i = P_ih is the directed distance from hyperplane h to point P_i.

Corollary 6B.    If the normalized ordinary hyperplanes h₁, h₂, h₃ are dependent and X is any point on the ideal line L through their normals, then

                          sin(a₂ - a₃)h₁ + sin(a₃ - a₁)h₂ + sin(a₁ - a₂)h₃ = 0

where a_i is the directed angle between X and h_i^N on L.

This can be proved by writing a dual of Theorem 7. The symbolic determinant will be: D = [ h₁, h₂, h₃; Xh₁, Xh₂, Xh₃;  Yh₁, Yh₂, Yh₃] where X and Y are arbitrary points. To prove Corollary 7B take X as chosen there, and point Y orthogonal to X on L (i.e. XY^t = 0, see F5).  Then Xh_i = cos a_i = d_i1 and Yh_i = sin a_i = d_i2 for i = 1, 2, 3. The first term of D will then be (d₂₁d₃₂ - d₂₂d₃₁)P₁ = [cos a₂ sin a₃ - sin a₂ cos a₃] P₁ = sin ( a₃ - a₂ ) P₁ and so on as in Corollary 3B, with a change of signs to get positive cycling of the subscripts.

When considering ideal points in Rⁿ and the angles between them, it may be more suitable to use vectors (with n - 1 components) and their dot products.

Corollary 6C.  Say the vectors V₁, V₂, V₃ are dependent.  Then c₁V₁ + c₂V₂ +  c₃V₃ = 0 where c₁ =  (d₂₁d₃₂ - d₂₂d₃₁), c2 =  (d₃₁d₁₂ - d₃₂d₁₁), c3 = (d₁₁d₂₂ - d₁₂d₂₁)  using d_ij = V_i .V_j (dot product).

This follows from the symbolic determinant D = [V₁, V₂, V₃; d₁₁,  d₂₁, d₃₁; d₁₂, d₂₂, d ₃₂].

12.   A perspective matrix

If a collineation f has a hyperplane has an axis it is called a perspective. From Theorem 5 we know f has a corresponding center point, and a matrix representation of the form T = I + hC where hyperplane h is the axis and C is the center. If point C is on h (Ch = 0), f is called an elation. If C is not on h then f is called an homology. Say P and Q are distinct points and Pf = Q. Then neither P nor Q are on h since f is one-to-one and all points on h are invariant. Now if Ph = Qh then f is an elation for PT = P + (Ph)C = Q implies Ph + (Ph)Ch = Qh which, using Ph = Qh, requires Ch = 0.  So if f is an homology, Ph /= Qh.

We will construct a matrix for an homology given its axis and two successive mappings. In Rⁿ the invariant axis of perspective f fixes the image of n - 1 independent points on the axis. When f is an homology, two additional points (off the axis) and their images will determine n + 1 maps, and hence f. But these points can not be chosen freely. For transforming P₁T = P₁( I + hC) = P₁ + (P₁h)C shows the transform of a point P₁ lies on the line through P₁ and C. It is also easy to show (VanArsdale) that if P₁T = P₂, T must be of the form

(A)                T = I + h[-P₁/(P₁h) + xP₂],   x any nonzero constant.

Now if we also transform P₂T = P₃, clearly the points P₁, P₂, P₃ must all lie on the same line through C. This dependence of three points provides an application for the explicit expressions of dependence in the previous section.

Theorem 8.   Say the homology f has hyperplane h as an axis, and for distinct ordinary points P₁, P₂, P₃:  P₁f = P₂ and P₂f = P₃ . Use the notation d_i = P_ih. Then f is represented by the matrix

              T = I + h( -P₁/d₁ + xP₂)      where x = [d₃(d₁ - d₂)] / [d₁d₂(d₂-d₃)].

Proof: Since P₁, P₂, P₃ are collinear we can apply Corollary 6A, using the axis h of  f as the arbitrary hyperplane in the corollary. So (d₂-d₃)P₁ + (d₃-d₁)P₂ + (d₁ - d₂)P₃ = 0, d_i as above.  Thus:

(B)         P₁ + [(d₃ - d₁)/(d₂ - d₃)] P₂ = kP₃

where k is a constant we need not be concerned with as long as it is not zero. Since f is an homology, from the above discussion we know d₁ /= d₂ and d₂ /= d₃.

Now applying  P₂T  in (A) gives P₂T = P₂ + d₂ [-P₁/d₁ + xP₂] or

(C)        P₂T = -(d₂/d₁)[P₁ - (d₁/d₂)(1 + d₂x)P₂]

We wish the expression in the brackets in (C) to equal a multiple of P₃ as in (B). So solving P₁ - (d₁/d₂)(1 + d₂x)P₂ =  P₁ + (d₃ - d₁)/(d₂ - d₃) P₂  for x gives x = d₃(d₁ - d₂)/d₁d₂(d₂-d₃) as in Theorem 8. Note that generally the homology of Theorem 8 will not be affine hence Theorem 3 above for affinities does not apply.

REFERENCES

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

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

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.

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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Corrections, references, comments or questions on this article are appreciated, but please no unrelated homework requests.
email Daniel W. VanArsdale:  barnowl@silcom.com

First uploaded Oct. 2, 2000. Sections 8-10 revised October 26, 2007. Hit counter added 11/15/2019.