Context: I am studying a graphics textbook which is analyzing something called a "3D projective transformation", which is neither linear nor affine but somehow preserves co-linearity and co-planarity:
Co-linearity means that, if three or more points are on a single line, the transformed points will also be on some single line. [Co-planarity has an analogous meaning.]
The textbook makes the claim:
As a result of [the preservation of co-planarity], we know that for points on a fixed triangle, we will have $z_n = ax_n + by_n + c$ for some fixed $a$, $b$, and $c$. Thus, the correct $z_n$ value for a point can be computed using linear interpoluation over the $2D$ image domain as long as we know its value at the three vertices of a triangle.
Here I am assuming that the book is saying that if $f$ is a 3D projective transformation, and if $a$, $b$, and $c$ are points in the domain of $f$. Then we can determine some point inside the triangle determined by these 3 points by the equation
$$ f(p) = (x_n, y_n, z_n) $$
such that $z_n = a x_n + b y_n + c$.
Am I getting this right? The textbook makes this point without any rigor. Is this some well known fact about triangles that somebody can point me to or explain?