Course MT3818 Topics in Geometry

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(Exercises 9)

Exercises 8

  1. Let f be the element f = (ax + b)/(cx - a) with a, bR, a2 + 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 xp, q is a point in RP1, prove that the cross-ratios ( p , q ; x , f(x) ) = ( p , q ; f(x) , x ) and deduce that these are -1.

    Solution to question 1

  2. 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.

    Solution to question 2

  3. Show that any four points (no three on a line) in RP2 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.]

    Solution to question 3

  4. 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 RP1, describe how to find a projective transformation of order 3 which maps a to b and b to c.

    Solution to question 4

  5. If a matrix AGL(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 Ta of PGL(2, R) which has ∞ as a unique fixed point and that restricted to the affine part of RP1 it represents a translation by a.
    Prove that any projective map with a single fixed point is conjugate to a map of the form Ta for some a.

    Solution to question 5

  6. 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.

    Solution to question 6

  7. 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.]

    Solution to question 7

  8. State the dual of Pappus's theorem.

    Solution to question 8

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JOC March 2003