# Fair surfaces - general mathematical theory

Fairness measures for surfaces are, in general, functionals containing more complicated terms thatn the usual bending energy, and may depend not only on the mean curvature but also on principal curvatures (weighted differently) and their derivatives. Something of the form:

$J(S)=\int_S L(k_1,k_2, \partial k_1,\partial k_2,\ldots)$

A fair surface is, therefore, a surface which is a critical point for one such generalized energy functional, the "easiest" example being minimal surfaces.

There is quite some literature on fair surfaces from the point of view of applications. In particular various constrcutions were developed on how to construct a fair surface from Bezier patches on a grid, trying then to smoothen this discrete approximation in a way that minimizes some fairness measure.

However I was unable to find references for the general theory of minimization of fairness functionals and I wonder if there is any relevant literature on the subject. In only know this PhD thesis http://www.cs.berkeley.edu/~sequin/PAPERS/Theses/Moreton_thesis.pdf

and wonder whether more recent material exists (possibily accesible by a good undergraduate). I am interested in existence results under various boundary conditions, maybe explicit solutions or general proprties of fair surfaces (or relations between different fairness measures).

• I'm not an expert, but I don't believe there is much of a general theory for minimizing such energies, especially ones involving derivatives of the principal curvatures (whose Euler-Lagrange equations are PDE's of order 6 or higher). As for energy functionals involving only the principal curvatures, one always restricts to convex surfaces, where the Euler-Lagrange equations are elliptic 4nd order PDE's. Even here, things are difficult, and usually special cases are studied. The main ones are the total mean curvature and the Willmore functional (I suggest googling these terms). – Deane Yang Aug 26 '15 at 13:20
• This survey cites 41 papers, although it is a decade old: Brook, Alexander, Alfred M. Bruckstein, and Ron Kimmel. "On similarity-invariant fairness measures." Scale Space and PDE Methods in Computer Vision. Springer Berlin Heidelberg, 2005. 456-467. – Joseph O'Rourke Aug 26 '15 at 14:19

Moreton's thesis work was included in a book that explores fairness measures in modeling from a variety of perspectives: Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-Aided Design (1994).

In addition to the chapter by Moreton and Séquin (focusing on curvature variation as a measure of fairness), there is a comprehensive chapter by Roulier and Rando that addresses the issue of how to mathematically define "fairness", comparing various fairness measures for curves and surface and offering implementation strategies.

A more recent collection of articles, concentrating on the implementation of a great variety of fairness measures, is Advanced Course on FAIRSHAPE (2012).