# Computing the Distance between Piecewise-Linear Bivariate Functions

* Auteur correspondant
1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : We consider the problem of computing the distance between two piecewise-linear bivariate functions $f$ and $g$ defined over a common domain $M$, induced by the $L_2$~norm, that is $\|f-g\|_2=\sqrt{\int_M (f-g)^2}$. 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 na\"ive algorithm requires $\Theta(n^2)$ arithmetic operations to compute this distance. We show that it is possible to compute it in $\O(n\log^4 n\log\log n)$ arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present several generalizations and an application to terrain matching.
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Type de document :
Article dans une revue
ACM Transactions on Algorithms, Association for Computing Machinery, 2016, 12 (1), pp.3:1-3:13
Domaine :

https://hal.archives-ouvertes.fr/hal-01112394
Contributeur : Guillaume Moroz <>
Soumis le : mardi 3 février 2015 - 11:02:15
Dernière modification le : jeudi 11 janvier 2018 - 06:25:24
Document(s) archivé(s) le : samedi 12 septembre 2015 - 07:40:23

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• HAL Id : hal-01112394, version 1

### Citation

Guillaume Moroz, Boris Aronov. Computing the Distance between Piecewise-Linear Bivariate Functions. ACM Transactions on Algorithms, Association for Computing Machinery, 2016, 12 (1), pp.3:1-3:13. 〈hal-01112394〉

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