Discrete Curvature

Theory and Applications


A colloquium on discrete curvature

Piecewise linear approximation of smooth functions of two variables

Joseph Fu

Slides are available by following the link.

Given a piecewise linear (PL) function p defined on an open subset U of euclidean space, one may construct by elementary means a unique polyhedron with multiplicities D(p) (the "gradient cycle" of p) in the cotangent bundle of U representing the graph of the differential of p. Restricting to dimension 2, we show that any smooth function f(x,y) may be approximated by a sequence of PL functions p such that the areas of the D(p) are locally dominated by the area of the graph of the differential of f times a universal constant. Using local 2nd order Taylor expansions of f(x,y), the main ingredients of the proof are an elementary analysis of the case where f(x,y) is quadratic, together with a well known mesh interpolation algorithm due to L.P. Chew.

This is joint work with Ryan Scott.

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