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- Let
*f*be the element*f*= (*ax*+*b*)/(*cx*-*a*) with*a*,*b*∈**R**,*a*^{2}+*bc*> 0. Find the order of*f*in the projective group*PGL*(2,**R**).

Prove that*f*has two fixed points,*p*,*q*equidistant from*f*(∞).

If*x*≠*p*,*q*is a point in**R***P*^{1}, prove that the cross-ratios (*p*,*q*;*x*,*f*(*x*) ) = (*p*,*q*;*f*(*x*) ,*x*) and deduce that these are -1. - Show that any three distinct points on a projective line can be mapped to any three distinct points on a second line by positioning the lines suitably in a plane and projecting from a point not on either line.
- Show that any four points (no three on a line) in
**R***P*^{2}can be mapped to any other four such points by a projective transformation.

[Hint: Show first that one can map [*α*, 0, 0], [0,*β*, 0] and [0, 0,*γ*] to three of the points. Then choose*α*,*β*,*γ*so that [1, 1, 1] is mapped to the fourth point.] - Write down a formula for a projective transformation which corresponds to the cyclic permutation of the set { ∞, 0, 1 }.

If*a*,*b*,*c*are any distinct points of**R***P*^{1}, describe how to find a projective transformation of order 3 which maps*a*to*b*and*b*to*c*. - If a matrix
*A*∈*GL*(2,**R**) represents an element*f*of*PGL*(2,**R**), prove that the eigenvectors of*A*correspond to the fixed points of*f*.

Show that the matrix with*a*≠ 0 represents an element*T*_{a}of*PGL*(2,**R**) which has ∞ as a unique fixed point and that restricted to the affine part of**R***P*^{1}it represents a translation by*a*.

Prove that any projective map with a single fixed point is conjugate to a map of the form*T*_{a}for some*a*.

- A
*pencil of lines*is a set of lines through a common point.

If a line*l*meets the pencil*OA*,*OB*,*OC*,*OD*as shown, show that the cross-ratio of the four points of intersection does not depend on which line*l*we take. - Prove Pappus's theorem (about 320 AD):

If the vertices of a hexagon lie alternately on two lines, then the meets of opposite sides are collinear.

[Hint: Project two of the meets to points on the line at infinity.] - State the dual of Pappus's theorem.

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