Stabbing Delaunay Tetrahedralizations |
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Authors: | Email author" target="_blank">Jonathan Richard?ShewchukEmail author |
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Institution: | (1) Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA 94720, USA |
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Abstract: | A Delaunay tetrahedralization of $n$ vertices is exhibited for which
a straight line can pass through the interiors of
$\Theta(n^2)$ tetrahedra.
This solves an open problem of Amenta,
who asked whether a line can stab more than $O(n)$ tetrahedra.
The construction generalizes to higher dimensions: in $d$ dimensions,
a line can stab the interiors of $\Theta(n^{\lceil d / 2 \rceil})$
Delaunay $d$-simplices.
The relationship between a Delaunay triangulation and
a convex polytope yields another result:
a two-dimensional slice of a $d$-dimensional $n$-vertex polytope
can have $\Theta(n^{\lfloor d / 2 \rfloor})$ facets.
This last result was first demonstrated by Amenta and Ziegler,
but the construction given here is simpler and more intuitive. |
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Keywords: | Delaunay triangulation Delaunay tetrahedralization |
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