The fact that a line in the projective plane has a homogeneous equation of the form a x + b y + c z = 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 = s a + t b 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 s a + t b 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 s a + t b 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}.
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} ?
Find a parametric representation for the line in P^{2} given by the homogeneous equation 6 x + 11 y + 9 z = 0.
Let f(x) = U x + v be the isometry of R^{2} given by
| U = |
| and v = |
| . |
Find the affine matrix of f relative to the affine basis
| { | (1,0), (0,1), (0,0) | } |
Find all fixed points of f.
Find all lines stabilized by f.
For each line stabilized by f provide the homogeneous equation for it as a line in P^{2}.
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 ?
In P^{2} should the line at infinity be regarded as a line that is stabilized by f ?
Let f(x) = U x + v be the order 2 affine transformation of R^{2} given by
| U = |
| and v = |
| . |
Find the affine matrix of f relative to the affine basis
| { | (1,0), (0,1), (0,0) | } |
Find all fixed points of f.
Find all lines stabilized by f.
For each line stabilized by f provide the homogeneous equation for it as a line in P^{2}.
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 ?
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 ?
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.