1,790 research outputs found
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
The Complexity of Finding Small Triangulations of Convex 3-Polytopes
The problem of finding a triangulation of a convex three-dimensional polytope
with few tetrahedra is proved to be NP-hard. We discuss other related
complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof
appeared at the proceedings of SODA 200
An Algorithm for Triangulating 3D Polygons
In this thesis, we present an algorithm for obtaining a triangulation of multiple, non-planar 3D polygons. The output minimizes additive weights, such as the total triangle areas or the total dihedral angles between adjacent triangles. Our algorithm generalizes a classical method for optimally triangulating a single polygon. The key novelty is a mechanism for avoiding non-manifold outputs for two and more input polygons without compromising opti- mality. For better performance on real-world data, we also propose an approximate solution by feeding the algorithm with a reduced set of triangles. In particular, we demonstrate experimentally that the triangles in the Delaunay tetrahedralization of the polygon vertices offer a reasonable trade off between performance and optimality
Flip-graph moduli spaces of filling surfaces
This paper is about the geometry of flip-graphs associated to triangulations
of surfaces. More precisely, we consider a topological surface with a
privileged boundary curve and study the spaces of its triangulations with n
vertices on the boundary curve. The surfaces we consider topologically fill
this boundary curve so we call them filling surfaces. The associated
flip-graphs are infinite whenever the mapping class group of the surface (the
group of self-homeomorphisms up to isotopy) is infinite, and we can obtain
moduli spaces of flip-graphs by considering the flip-graphs up to the action of
the mapping class group. This always results in finite graphs and we are
interested in their geometry.
Our main focus is on the diameter growth of these graphs as n increases. We
obtain general estimates that hold for all topological types of filling
surface. We find more precise estimates for certain families of filling
surfaces and obtain asymptotic growth results for several of them. In
particular, we find the exact diameter of modular flip-graphs when the filling
surface is a cylinder with a single vertex on the non-privileged boundary
curve.Comment: 52 pages, 29 figure
Triangulating a Cappell-Shaneson knot complement
We show that one of the Cappell-Shaneson knot complements admits an
extraordinarily small triangulation, containing only two 4-dimensional
simplices.Comment: 9 pages, 5 figures. V2->V3: Primary figures now display a symmetry of
order two. Attaching maps made "order-preserving". A new figure describing
the dual 2-cells is included. Previous arguments simplified a little, due to
the symmetr
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