# Polygon with curved sides, and higher-dimensional generalizations

I am trying to find references about generalizations of polygons with non-straight sides.

I am interested in both the convex and non-convex cases, and particularly in polynomial boundaries, and algorithms to compute their area. References about higher-dimensional analogues (bodies with piecewise polynomial boundaries) and numerical methods to calculate more general functions over such sets would also be highly welcome.

So far, I have been unable to find much myself, but that might be because I am not aware of the correct search terms.

Questions: What are some references and good search terms for the study of non-/convex polygons, -hedra, -topes with curved, especially polynomial, boundaries?

• You might have a look at semialgebraic sets and subanalytic sets. – Andrew D. Hwang Apr 9 '16 at 18:14
• I know that Stasheff originally realized the associahedron as what Wikipedia calls a "curvilinear polytope." A cursory search for this term doesn't seem to yield anything too promising. – pjs36 Apr 9 '16 at 19:46
• Thank you. @AndrewD.Hwang, if you convert your comment to an answer I will be happy to accept it. – Eckhard Apr 10 '16 at 6:47
• @Eckhard: Done. :) If I knew more about the area, I'd have happily added more exposition. The Real Algebraic and Analytic Geometry preprint server might be a productive source for "hard details".... – Andrew D. Hwang Apr 10 '16 at 15:28

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Cpx}{\mathbf{C}}$Just a few search terms:

• A semialgebraic set is a finite union of subsets of $\Reals^{n}$, each defined by finitely many polynomial equalities and/or strict inequalities.

• A subanalytic set in $\Reals^{n}$ is locally a union of sets defined by finitely many real-analytic inequalities.

• An analytic polyhedron is a subset of $\Cpx^{n}$ defined by finitely many inequalities $|f(z)| < 1$, with $f$ holomorphic.

Examples include graphs of polynomial functions and relations, a region bounded by such a graph, and the like. Open and closed half-spaces are semialgebraic, so ordinary polyhedra (which are suitable finite intersections of half-spaces) fall under this umbrella.

In Differential Geometry textbook by HW Guggenheimer it is mentioned among exercises $Liouville$ polar Curves $r^n = \cos n \theta$ for integer $n$ values. Among them are hyperbolae (equilateral), straight line, circles one through origin and one centered around origin, Lemniscate of Bernoulli etc.

Radius vector makes an angle to curve

$$= n \theta + ( 2 k -1) \pi/2.$$