214 research outputs found
Triangulations of nearly convex polygons
Counting Euclidean triangulations with vertices in a finite set \C of the
convex hull \conv(\C) of \C is difficult in general, both algorithmically
and theoretically. The aim of this paper is to describe nearly convex polygons,
a class of configurations for which this problem can be solved to some extent.
Loosely speaking, a nearly convex polygon is an infinitesimal perturbation of a
weakly convex polygon (a convex polygon with edges subdivided by additional
points). Our main result shows that the triangulation polynomial, enumerating
all triangulations of a nearly convex polygon, is defined in a straightforward
way in terms of polynomials associated to the ``perturbed'' edges
Triangulations of hyperbolic 3-manifolds admitting strict angle structures
It is conjectured that every cusped hyperbolic 3-manifold has a decomposition
into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation
of the manifold). Under a mild homology assumption on the manifold we construct
topological ideal triangulations which admit a strict angle structure, which is
a necessary condition for the triangulation to be geometric. In particular,
every knot or link complement in the 3-sphere has such a triangulation. We also
give an example of a triangulation without a strict angle structure, where the
obstruction is related to the homology hypothesis, and an example illustrating
that the triangulations produced using our methods are not generally geometric.Comment: 28 pages, 9 figures. Minor edits and clarification based on referee's
comments. Corrected proof of Lemma 7.4. To appear in the Journal of Topolog
The Number of Triangles Needed to Span a Polygon Embedded in R^d
Given a closed polygon P having n edges, embedded in R^d, we give upper and
lower bounds for the minimal number of triangles t needed to form a
triangulated PL surface in R^d having P as its geometric boundary. The most
interesting case is dimension 3, where the polygon may be knotted. We use the
Seifert suface construction to show there always exists an embedded surface
requiring at most 7n^2 triangles. We complement this result by showing there
are polygons in R^3 for which any embedded surface requires at least 1/2n^2 -
O(n) triangles. In dimension 2 only n-2 triangles are needed, and in dimensions
5 or more there exists an embedded surface requiring at most n triangles. In
dimension 4 we obtain a partial answer, with an O(n^2) upper bound for embedded
surfaces, and a construction of an immersed disk requiring at most 3n
triangles. These results can be interpreted as giving qualitiative discrete
analogues of the isoperimetric inequality for piecewise linear manifolds.Comment: 16 pages, 4 figures. This paper is a retitled, revised version of
math.GT/020217
Few smooth d-polytopes with n lattice points
We prove that, for fixed n there exist only finitely many embeddings of
Q-factorial toric varieties X into P^n that are induced by a complete linear
system. The proof is based on a combinatorial result that for fixed nonnegative
integers d and n, there are only finitely many smooth d-polytopes with n
lattice points. We also enumerate all smooth 3-polytopes with at most 12
lattice points. In fact, it is sufficient to bound the singularities and the
number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result
Bounds for the genus of a normal surface
This paper gives sharp linear bounds on the genus of a normal surface in a
triangulated compact, orientable 3--manifold in terms of the quadrilaterals in
its cell decomposition---different bounds arise from varying hypotheses on the
surface or triangulation. Two applications of these bounds are given. First,
the minimal triangulations of the product of a closed surface and the closed
interval are determined. Second, an alternative approach to the realisation
problem using normal surface theory is shown to be less powerful than its dual
method using subcomplexes of polytopes.Comment: 38 pages, 25 figure
Analysis and new constructions of generalized barycentric coordinates in 2D
Different coordinate systems allow to uniquely determine the position of a geometric element in space. In this dissertation, we consider a coordinate system that lets us determine the position of a two-dimensional point in the plane with respect to an arbitrary simple polygon. Coordinates of this system are called generalized barycentric coordinates in 2D and are widely used in computer graphics and computational mechanics. There exist many coordinate functions that satisfy all the basic properties of barycentric coordinates, but they differ by a number of other properties. We start by providing an extensive comparison of all existing coordinate functions and pointing out which important properties of generalized barycentric coordinates are not satisfied by these functions. This comparison shows that not all of existing coordinates have fully investigated properties, and we complete such a theoretical analysis for a particular one-parameter family of generalized barycentric coordinates for strictly convex polygons. We also perform numerical analysis of this family and show how to avoid computational instabilities near the polygon’s boundary when computing these coordinates in practice. We conclude this analysis by implementing some members of this family in the Computational Geometry Algorithm Library. In the second half of this dissertation, we present a few novel constructions of non-negative and smooth generalized barycentric coordinates defined over any simple polygon. In this context, we show that new coordinates with improved properties can be obtained by taking convex combinations of already existing coordinate functions and we give two examples of how to use such convex combinations for polygons without and with interior points. These new constructions have many attractive properties and perform better than other coordinates in interpolation and image deformation applications
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