I can "almost" parameterize the boundary of a square using $${\bf r}(t) = (\cos t)^{1/p} {\bf i} + (\sin t)^{1/p} {\bf j},$$ $0\leq t\leq 2 \pi$, and $p$ is odd. This parameterization is smooth (or at least $C^1$), and of course is the unit ball in the $L^p$ norm. Letting $p\rightarrow \infty$ makes our approximation better.
Now suppose I have a triangle, or in general, a convex polygon with vertices $a_1, a_2, \dots, a_n$. Is there some relatively simple, explicit smooth approximate parameterization of the boundary? It should, of course, have some tweakable parameter that allows for convergence, like $p$ in the example I gave. The simpler, and the "cuter", the better.
Another issue is the following. With my parameterization of the square, if I want to get a list of roughly equally distributed points on the boundary, I cannot do this by letting $t=2 \pi i/N$. In that case the points collect around the corners on the square. Is there some nice way of fixing this for the square, and for my polygon in general?
An arc-length parametrization of your polygon uses piecewise-linear functions: if $a_j = (b_j, c_j)$, $r(t) = \sum_j u_j(t) a_j$ where $u_j(t)$ is a "triangular" function of the form $$ u_j(t) = \cases{ \dfrac{t-t_{j-1}}{t_j - t_{j-1}} & for $t_{j-1} \le t \le t_j$\cr \dfrac{t_{j+1} - t}{t_{j+1}-t_j} & for $t_j \le t \le t_{j+1}$\cr 0 & otherwise}$$ You can smooth it out e.g. by replacing $u_j(t)$ by $$\frac{p}{\pi} \int_{-\infty}^\infty \frac{u_j(s)}{p^2 + (t-s)^2}\ ds$$ for $p \to 0+$. This is a bit messy, but can be written explicitly using logarithms and arctan.