Transformation Geometry — Math 331

March 26, 2004

Projective Transformations of the Projective Plane

An affine transformation of the affine plane R^{2}, via its affine matrix relative to a given affine basis, gives rise to a transformation of the projective plane P^{2}. Because the affine matrix is an invertible 3 × 3 matrix, it gives rise to an invertible linear transformation of R^{3} and, therefore, an invertible map from P^{2} to itself. Such a map on P^{2} is an example of a projective transformation.

Definition. A projective transformation of P^{2} is an invertible map f from P^{2} to itself arising from some invertible 3 × 3 matrix M in the sense that if a point p of P^{2} is represented by a vector x of homogeneous coordinates in R^{3}, then the point f(p) in P^{2} is the point for which the vector Mx (with x regarded as a length 3 column) is a triple of homogeneous coordinates.

It is obvious that if both x and x′ are vectors of homogeneous coordinates for a point p in P^{2}, then Mx and Mx′ represent the same point q in P^{2}. Morever, if M′ = bM for some non-zero scalar b ≠ 0, then M′x = (bM)x = b(Mx), which is to say that if M′ is a non-zero scalar multiple of M, then M and M′ determine the same projective transformation f.

Proposition. An invertible 3 × 3 matrix M is the affine matrix of an affine transformation of R^{2} relative to a given affine basis of R^{2} if and only if each of its columns sums to the number 1.

Proof. By definition of the affine matrix of an affine transformation each column is a triple of barycentric coordinates and, hence, sums to 1. Conversely, given such an invertible matrix there is a unique affine transformation sending the members of the affine basis of R^{2} to the three barycentric combinations of the three points in the basis represented by the columns of the given matrix.

Theorem. A point p in P^{2} is a fixed point of the projective transformation f given by an invertible 3 × 3 matrix M if and only if a vector x ≠ 0 of homogeneous coordinates of p is an eigenvector of M.

Proof. Regardless of the value of a non-zero scalar λ, the condition Mx = λx is precisely equivalent to f(p) = p.

Theorem. A line with homogeneous equation a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} = 0 (where a = (a_{1}, a_{2}, a_{3}) ≠ (0, 0, 0)) in P^{2} is stabilized by the projective transformation f given by an invertible 3 × 3 matrix M if and only if its coefficient vector a is an eigenvector of the transpose of M.

Proof. Let l be the line. The point with homogeneous coordinate vector x lies on f^{−1}(l) if and only if f(x) lies on l if and only if ^{t}a (Mx) = 0. Since ^{t}a (Mx) = (^{t}aM) x = ^{t} (^{t}Ma)x, the line f^{−1}(l) has the equation b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} = 0 where b = ^{t}Ma, and, therefore, f(l) = l if and only if f^{−1}(l) = l if and only if a and b are parallel vectors, i.e., a is an eigenvector of ^{t}M.

Corollary. The line at infinity, i.e., the line x_{1} + x_{2} + x_{3} = 0, is stabilized by the projective transformation given by M if and only if M is a non-zero scalar multiple of the affine matrix of an affine transformation.

Proof. By the theorem the line at infinity is stabilized if and only if the coefficient vector (1, 1, 1) is an eigenvector of ^{t}M. If λ ≠ 0 is the corresponding eigenvalue of ^{t}M, then this condition is equivalent to the statement that each column of λ^{−1}M sums to 1.

Exercises due Monday, March 29

1. Find the affine matrix of the translation of R^{2} by the vector (3, 4), and then find all fixed points and all stabilized lines of the corresponding projective transformation.

2. Repeat the previous exercise for the glide reflection f of R^{2} given by

f(x) = \frac{1}{13} ( \begin{matrix}12 & 5 \\ 5 & −12 \end{matrix} ) ( \begin{matrix}x_{1} \\ x_{2} \end{matrix} ) + ( \begin{matrix}6 \\ − 4 \end{matrix} ) .

3. Give an example of an invertible 3 × 3 matrix for which the corresponding projective transformation carries the line at infinity to the line z = 0.

4. What specific restriction on a 3 × 3 matrix is imposed by the condition that it stabilizes the line in P^{2} that has the homogeneous equation z = 0?

5. Why must every projective transformation of P^{2} have at least one fixed point and one stabilized line?