# Superposed Ranges

By Axiom, a projectivity relating two ranges on one line cannot have more than two invariant points without being the identity. Transformations classified by invariant points Elliptic - 0 invariant points Parabolic - 1 invariant point Hyperbolic - 2 invariant points
Construct a hyperbolic or parabolic projectivity: Consider five collinear points G,H,I,K,L. HLJ maps through D to CBJ maps through A to IGJ. J is invariant Any other invariant point must be collinear with the two projective centers, D and A. That can only be K. The projectivity HLJ to IGJ is hyperbolic if J and K are distinct, parabolic if J and K coincide.