Superposed Ranges
- Author:
- GeoGebra Materials Team
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.