Say P ~ Q (P and Q are «projectively equivalent») iff there is a projective transformation f such that f(P) = Q. Then ~ is an equivalence relation. I read that the space of inscribed n-gons modulo projective equivalence has dimension n-3. Why is this? Also, are there any related results?
Of the projective transformations fixing a conic, there is a unique one sending any given ordered triple of points to any other given ordered triple of points. So you are free to determine the location of the first 3 vertices of the n-gon, modulo projective equivalence, but any two placements of the remaining (n-3) points are projectively inequivalent.
Thus, the space of n-gons up to projective equivalence can be thought of as the space of (n-3) points on a projective line (or conic, it is isomorphic) punctured at three given points. This has dimension (n-3).