Projective Collineations in Homogeneous Coordinates

Let us remind ourselves about incidence and collinearity in homogeneous coordinates. (1) A point and a line are incident if the inner product of their vectors is 0 (2) Three points are collinear if the matrix formed by their vectors has determinant 0 (3) Three points are non-collinear if the matrix formed by their vectors has non-zero determinant So what is a projective collineation? It is a bijective mapping that sends collinear points to collinear points. One mapping that we studied extensively in linear algebra is the linear transformation, which has a matrix representation. Could this be a projective collineation? 1. Show that a non-singular (invertible) matrix is a bijective mapping that sends collinear points to collinear points. 2. Show that any bijective mapping that sends collinear points to collinear points must be a bijective linear transformation. Conclude that projective collineations are exactly invertible matrices. 3. Can two different invertible matrices give the same projective collineation? Give an example. This suddenly makes projective collineations more tangible, and perhaps easier to understand. 4. Use homogeneous coordinates to show that projective collineations preserve incidence. Recall the Fundamental Theorem of Projective Geometry. Fundamental Theorem of Projective Geometry Let ABCD be four points such that no three are collinear. Then the mapping ABCD −→ A0B0C0D0 completely determines a projective collineation. 5. Why four and not three? Come up with an example of two different matrices that both send (1,0,0) to itself, (0,1,0) to itself and (0,0,1) to itself, but one sends (1,1,1) to itself but the other sends (1,1,1) to (1,1,2). FTPG Corollary A projective collineation which fixes four points is the identity. 6. Explain why the corollary is true, given FTPG. 7. Use homogeneous coordinates to show that projective collineations will always fix a point and a line. This may involve eigenvectors/eigenvalues. Knowing the eigenvector/eigenvalue structure of the matrix can actually help us determine the number of fixed points and lines. Type I: All points are fixed. Type II: There is only one linewise fixed point and one pointwise fixed line, which are incident. Type III: There is only one linewise fixed point and one pointwise fixed line, which are not incident. Type IV: There is only one fixed point and one fixed line, which are incident.

1 Type V: There is only one fixed point and one fixed line, which are not incident. Type VI: There are only two fixed points and two fixed lines, such that one fixed point is incident with both fixed lines and one fixed line is incident with both fixed points. Type VII: There are only three fixed points and three fixed lines, which form a fixed triangle. 8. For each type, determine the eigenvector/eigenvalue structure. 9. Categorize the following into types: translation, shear, dilation, reflection, 180 degree rota- tion, 90 degree rotation, stretch. 10. Find an example of another projective collineation we have not yet mentioned. Draw this, and identify its type by finding its fixed points and lines.

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