ABSTRACT
Until now, our study of curves and surfaces has always described them as subsets of some ambient Euclidean space Rn. We defined parametrizations as vector functions of one (for a curve) or two (for a surface) variables into R2 or R3, without pointing out that many of our constructions relied on the fact that R2 and R3 are topological vector spaces. That we have only studied geometric objects that are subsets of R3 does not bely our intuition since the daily reality of human experience evolves (or at least appears to evolve) completely in three dimensions that we feel are flat. However, both in mathematics and in physics, one does not need to take such a large step in abstraction to realize the insufficiency of this intuition.
