# Computing the Distance between Piecewise-Linear Bivariate Functions

1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : We consider the problem of computing the distance between two piecewise-linear bivariate functions $f$ and $g$ defined over a common domain $M$. We focus on the distance induced by the $L_2$-norm, that is $\|f-g\|_2=\sqrt{\iint_M (f-g)^2}$. If $f$ is defined by linear interpolation over a triangulation of $M$ with $n$ triangles, while $g$ is defined over another such triangulation, the obvious naïve algorithm requires $\Theta(n^2)$ arithmetic operations to compute this distance. We show that it is possible to compute it in $\O(n\log^4 n)$ arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.
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https://hal.inria.fr/inria-00608255
Contributor : Guillaume Moroz <>
Submitted on : Tuesday, July 12, 2011 - 10:58:11 PM
Last modification on : Wednesday, August 14, 2019 - 10:46:02 AM
Long-term archiving on : Monday, December 5, 2016 - 12:28:45 AM

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### Identifiers

• HAL Id : inria-00608255, version 2
• ARXIV : 1107.2312

### Citation

Guillaume Moroz, Boris Aronov. Computing the Distance between Piecewise-Linear Bivariate Functions. SODA - Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms - 2012, Jan 2012, Kyoto, Japan. ⟨inria-00608255v2⟩

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