Transformation Geometry — Math 331

March 17, 2004

Parametric Form of a Line in the Projective Plane

The fact that a line in the projective plane has a homogeneous equation of the form ax + by + cz = 0 (where (a, b, c) ≠ (0, 0, 0)) reflects fact that P^{2} has one point for each line through the origin in R^{3} and a line in P^{2} consists of the points in P^{2} corresponding to lines in R^{3} lying in a plane through the origin of R^{3}. Just as a plane through the origin in R^{3}, which is the same thing as a 2-dimensional linear subspace of R^{3}, consists of the set of all linear combinations c = sa + tb of two linearly independent vectors a, b in R^{3}, i.e., a linear basis of the plane, the line in P^{2} through two different points a, b may be represented as the set of all linear combinations sa + tb of homogeneous coordinate vectors in R^{3} for the two given points with (s, t) ≠ (0, 0). To view a line in P^{2} as the set of all linear combinations of two of its points a and b is to provide what is sometimes called a parametric representation of the line — with parameters s and t.

Note that every line in P^{2} other than the line at infinity (the line x + y + z = 0) meets the line at infinity in a single point. If neither of two points a and b lies on the line at infinity, then homogeneous coordinate vectors for those points may be chosen so that a_{1} + a_{2} + a_{3} = 1 and b_{1} + b_{2} + b_{3} = 1, and then each of the points on the line through a and b except its single point on the line at infinity may be represented as sa + tb with s + t = 1, consistent with the fact that these points all correspond to the barycentric combinations of the points (a_{1}, a_{2}) and (b_{1}, b_{2}) in R^{2}.

Exercises due Friday, March 19

1. Find a homogeneous equation for the line in P^{2} containing the points of P^{2} with homogeneous coordinates (1, −2, 2) and (2, −1, −1). What is the ordinary equation for this line as a line in R^{2}?

2. Find a parametric representation for the line in P^{2} given by the homogeneous equation 6x + 11y + 9z = 0.

3. Let f(x) = Ux + v be the isometry of R^{2} given by

U = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) and v = ( \begin{matrix}−3 \\ 4 \end{matrix} ) .

(a) Find the affine matrix of f relative to the affine basis {(1, 0), (0, 1), (0, 0)} of R^{2}.

(b) Find all fixed points of f.

(c) Find all lines stabilized by f.

(d) For each line stabilized by f provide the homogeneous equation for it as a line in P^{2}.

(e) How may one see that a line in P^{2} is a stabilized line for f by considering its homogeneous equation in relation to the affine matrix of f?

(f) In P^{2} should the line at infinity be regarded as a line that is stabilized by f?

4. Let f(x) = Ux + v be the order 2 affine transformation of R^{2} given by

U = ( \begin{matrix}1 & 3 \\ 0 & −1 \end{matrix} ) and v = ( \begin{matrix}−3 \\ 2 \end{matrix} ) .

(a) Find the affine matrix of f relative to the affine basis {(1, 0), (0, 1), (0, 0)} of R^{2}.

(b) Find all fixed points of f.

(c) Find all lines stabilized by f.

(d) For each line stabilized by f provide the homogeneous equation for it as a line in P^{2}.

(e) How may one see that a point in P^{2} is fixed by f from considering its homogeneous coordinate vector in relation to the affine matrix of f?

(f) How may one see that a line in P^{2} is a stabilized line for f by considering its homogeneous equation in relation to the affine matrix of f?

5. Show by techniques of linear algebra that the non-orthogonal matrix U of the preceding problem is similar (under conjugation by an invertible matrix) to the orthogonal matrix U of the problem before it. Then explain why the affine transformation of the preceding problem cannot be conjugate to the affine transformation of the problem before it.