Deduce that on the projective line, all involutions with one fixed point have two fixed points, and thence that they map points to their harmonic conjugates with respect to these two fixed points. Let a collineation H be represented by a matrix as in exercise 3.1.1 . Show that H always has two distinct fixed points, either both real or both complex. Hx , then . Given the invariance of the cross ratio under complex collineations, show that all involutions of the projective line map points to their harmonic conjugates with respect to the two fixed points of the involution.
Involutions are useful when dealing with images of symmetrical objects.
Consider the simple case of a planar object with reflection symmetry, as in fig. In 3D, the lines joining corresponding points of the object are parallel. In the image, all such lines meet at the projection of the associated point at infinity S . The projection of the line of symmetry consists of points conjugate to S with respect of pairs of corresponding points on the shape. There is a plane involution of the image mapping points on the object to their opposites, and fixing S and each point of the symmetry axis.