28,555 research outputs found
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
Volume entropy for surface groups via Bowen-Series like maps
We define a Bowen-Series like map for every geometric presentation of a
co-compact surface group and we prove that the volume entropy of the
presentation is the topological entropy of this particular (circle) map.
Finally we find the minimal volume entropy among geometric presentations
Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization
In this paper, we propose a general framework for constructing IGA-suitable
planar B-spline parameterizations from given complex CAD boundaries consisting
of a set of B-spline curves. Instead of forming the computational domain by a
simple boundary, planar domains with high genus and more complex boundary
curves are considered. Firstly, some pre-processing operations including
B\'ezier extraction and subdivision are performed on each boundary curve in
order to generate a high-quality planar parameterization; then a robust planar
domain partition framework is proposed to construct high-quality patch-meshing
results with few singularities from the discrete boundary formed by connecting
the end points of the resulting boundary segments. After the topology
information generation of quadrilateral decomposition, the optimal placement of
interior B\'ezier curves corresponding to the interior edges of the
quadrangulation is constructed by a global optimization method to achieve a
patch-partition with high quality. Finally, after the imposition of
C1=G1-continuity constraints on the interface of neighboring B\'ezier patches
with respect to each quad in the quadrangulation, the high-quality B\'ezier
patch parameterization is obtained by a C1-constrained local optimization
method to achieve uniform and orthogonal iso-parametric structures while
keeping the continuity conditions between patches. The efficiency and
robustness of the proposed method are demonstrated by several examples which
are compared to results obtained by the skeleton-based parameterization
approach
JSJ-decompositions of finitely presented groups and complexes of groups
A JSJ-splitting of a group over a certain class of subgroups is a graph
of groups decomposition of which describes all possible decompositions of
as an amalgamated product or an HNN extension over subgroups lying in the
given class. Such decompositions originated in 3-manifold topology. In this
paper we generalize the JSJ-splitting constructions of Sela, Rips-Sela and
Dunwoody-Sageev and we construct a JSJ-splitting for any finitely presented
group with respect to the class of all slender subgroups along which the group
splits. Our approach relies on Haefliger's theory of group actions on CAT
spaces
Constructing Simplicial Branched Covers
Branched covers are applied frequently in topology - most prominently in the
construction of closed oriented PL d-manifolds. In particular, strong bounds
for the number of sheets and the topology of the branching set are known for
dimension d<=4. On the other hand, Izmestiev and Joswig described how to obtain
a simplicial covering space (the partial unfolding) of a given simplicial
complex, thus obtaining a simplicial branched cover [Adv. Geom. 3(2):191-255,
2003]. We present a large class of branched covers which can be constructed via
the partial unfolding. In particular, for d<=4 every closed oriented PL
d-manifold is the partial unfolding of some polytopal d-sphere.Comment: 15 pages, 8 figures, typos corrected and conjecture adde
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