In chapter 4 of "Curves and surfaces for CAGD" by Gerald Farin it is asked to find a curve scheme (i understood it as a procedure to produce a curve since the topic was De Casteljau algorithm) which is not affine invariant (a property of Bézier Curves).
I tried a few examples but i didn't come up with a single curve (or procedure) which is affine invariant. In reality I don't even understand what affine invariance is to a random curve or scheme that produces a curve. For example a circumference might be considered affine invariant if we are classifying conics under an affine point of view (the class of ellipses).
First let’s understand affine invariance. Suppose we have a curve construction scheme that produces a curve $C[d_1,\ldots,d_n](t)$ from input data $d_1, \ldots, d_n$. The construction is said to be affine invariant if for any affine transformation $T$ we have $$ T(C[d_1,\ldots,d_n](t)) = C[T(d_1),\ldots,T(d_n](t) $$ In words, you get the same result from two different calculations:
An example of a curve scheme that’s not affinely invariant is the “circle through three points” construction. If you create a circle through three points and then affinely transform it, you (generally) get an ellipse. But if you transform the three points and then construct a circle through them, you always get a circle, obviously.