**Application Challenges to Computational Geometry**.

The Computational Geometry Impact Task Force Report.

Tech. Rep. TR-521-96, Princeton University, April 1996.

Advances in Discrete and Computational Geometry – Proc. 1996 AMS-IMS-SIAM Joint Summer Research Conf. Discrete and Computational Geometry: Ten Years Later, Contemporary Mathematics 223, Amer. Math. Soc., 1999, pp. 407–423.**Optimal point placement for mesh smoothing**.

N. Amenta, M. Bern, and D. Eppstein.

*8th ACM-SIAM Symp. Discrete Algorithms,*New Orleans, 1997, pp. 528–537.

Symp. Computational Geometry Approaches to Mesh Generation, SIAM 45th Anniversary Mtg., Stanford, 1997.

arXiv:cs.CG/9809081.

*J. Algorithms*30: 302–322, 1999 (special issue for SODA 1997).We study finite element mesh smoothing problems in which we move vertex locations to optimize the shapes of nearby triangles. Many such problems can be solved in linear time using generalized linear programming; we also give efficient algorithms for some non-LP-type mesh smoothing problems. One lemma may be of independent interest: the locus of points in R

^{d}from which a d-1 dimensional convex set subtends a given solid angle is convex.(BibTeX – SODA paper – Citations – CiteSeer – ACM DL)

**The crust and the beta-skeleton: combinatorial curve reconstruction**.

N. Amenta, M. Bern, and D. Eppstein.

*Graphical Models & Image Processing*60/2 (2): 125–135, 1998.We consider the problem of "connect the dots": if we have an unknown smooth curve from which sample points have been selected, we would like to find a curve through the sample points that approximates the unknown curve. We show that if the local sample density is sufficiently high, a simple algorithm suffices: form the Delaunay triangulation of the sample points together with their Voronoi vertices, and keep only those Delaunay edges connecting original sample points. There have been many follow-up papers suggesting alternative methods, generalizing the problem to the reconstruction of curves with sharp corners or to curves and surfaces in higher dimensions, etc.

(BibTeX – Citations – CiteSeer – ACM DL)

**Regression depth and center points**.

N. Amenta, M. Bern, D. Eppstein, and S.-H. Teng.

arXiv:cs.CG/9809037.

*3rd CGC Worksh. Computational Geometry*, Brown Univ., 1998.

*Disc. Comp. Geom.*23 (3): 305–323, 2000.We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least ceiling(n/(d+1)). as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems.

(BibTeX – Citations – CiteSeer)

**Emerging challenges in computational topology**.

M. Bern, D. Eppstein, et al.

arXiv:cs.CG/9909001.

This is the report from the ACM Workshop on Computational Topology run by Marshall and myself in Miami Beach, June 1999. It details goals, current research, and recommendations in this emerging area of collaboration between computer science and mathematics.

(BibTeX – Citations – CiteSeer)

Co-authors – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine

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