This question refers to the last part of Chapter 1, Volume 2, of Spivak's Comprehensive Introduction to Differential Geometry (roughly corresponding to pp. 36-48).
It is known that, for planar curves, the only curves of constant affine curvature, up to a transformation in $SL(2,\mathbb{R})$, are the conic sections.
What about curves of constant affine curvature in $\mathbb{R}^n$?
I believe this corresponds to paths $[a,b] \to SL(n,\mathbb{R})$ but I am not 100% certain.
A pointer to a reference will suffice for an answer.
My guess (attempting to generalize from the case $n=2$) would be that they correspond either to: (1) conic sections embedded in $\mathbb{R}^n$, (2) curves defined by polynomials of degree $\le n$.