Bob Fisher
A coplanarity invariant on 5 corresponding points can be developed based on the cross ratio.
Assume that we have 5 coplanar points { a, b, c, d, e } in general position. From these, construct four additional points { u, v, x, y }, as shown in:
The constructed points define two lines with four colinear points, from which these cross ratio invariants can be defined:
These values are invariant if and only if the five points are coplanar.