I want to prove that a Rational Bézier Curve is not only affine invariant but also a projective invariant. By affine invariance i mean that applying an affine map to the curve is the same as applying it to it's control points and then building the curve from their images. When books like CAGED and NURBS talk about projective invariance i find them not so formal. What do they mean by projective map? For me a projective map is a map between projective spaces induced to quotients by a linear injective map between vector spaces. Meanwhile a Rational Bézier Curve lives in $\mathbb{R}^3$.
Edit: Should i consider $\mathbb{R}^3$ as a subset of $\mathbb{RP}^3$ with the map that sends $(x,y,z) \to [x,y,z,1]$? Applying a projective map to a point in $\mathbb{R}^3$ means applying it to $[x,y,z,1]$ and then projecting it dividing by the forth homogeneous coordinate? How can i apply a projective map (between projective spaces) to something living in a vector space...