2,531 research outputs found
Volumetric Untrimming: Precise decomposition of trimmed trivariates into tensor products
3D objects, modeled using Computer Aided Geometric Design tools, are
traditionally represented using a boundary representation (B-rep), and
typically use spline functions to parameterize these boundary surfaces.
However, recent development in physical analysis, in isogeometric analysis
(IGA) in specific, necessitates a volumetric parametrization of the interior of
the object. IGA is performed directly by integrating over the spline spaces of
the volumetric spline representation of the object. Typically, tensor-product
B-spline trivariates are used to parameterize the volumetric domain. A general
3D object, that can be modeled in contemporary B-rep CAD tools, is typically
represented using trimmed B-spline surfaces. In order to capture the generality
of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi
and Elber (2016) proposed the use of trimmed trivariates volumetric elements.
However, the use of trimmed geometry makes the integration process more
difficult since integration over trimmed B-spline basis functions is a highly
challenging task. In this work, we propose an algorithm that precisely
decomposes a trimmed B-spline trivariate into a set of (singular only on the
boundary) tensor-product B-spline trivariates, that can be utilized to simplify
the integration process in IGA. The trimmed B-spline trivariate is first
subdivided into a set of trimmed B\'ezier trivariates, at all its internal
knots. Then, each trimmed B\'ezier trivariate, is decomposed into a set of
mutually exclusive tensor-product B-spline trivariates, that precisely cover
the entire trimmed domain. This process, denoted untrimming, can be performed
in either the Euclidean space or the parametric space of the trivariate. We
present examples on complex trimmed trivariates' based geometry, and we
demonstrate the effectiveness of the method by applying IGA over the
(untrimmed) results.Comment: 18 pages, 32 figures. Contribution accepted in International
Conference on Geometric Modeling and Processing (GMP 2019
Asymptotically efficient triangulations of the d-cube
Let and be polytopes, the first of "low" dimension and the second of
"high" dimension. We show how to triangulate the product
efficiently (i.e., with few simplices) starting with a given triangulation of
. Our method has a computational part, where we need to compute an efficient
triangulation of , for a (small) natural number of our
choice. denotes the -simplex.
Our procedure can be applied to obtain (asymptotically) efficient
triangulations of the cube : We decompose , for
a small . Then we recursively assume we have obtained an efficient
triangulation of the second factor and use our method to triangulate the
product. The outcome is that using and , we can triangulate
with simplices, instead of the achievable
before.Comment: 19 pages, 6 figures. Only minor changes from previous versions, some
suggested by anonymous referees. Paper accepted in "Discrete and
Computational Geometry
Isotopic Equivalence from Bezier Curve Subdivision
We prove that the control polygon of a Bezier curve B becomes homeomorphic
and ambient isotopic to B via subdivision, and we provide closed-form formulas
to compute the number of iterations to ensure these topological
characteristics. We first show that the exterior angles of control polygons
converge exponentially to zero under subdivision.Comment: arXiv admin note: substantial text overlap with arXiv:1211.035
Asymptotic enumeration of non-crossing partitions on surfaces
We generalize the notion of non-crossing partition on a disk to general surfaces
with boundary. For this, we consider a surface S and introduce the number CS(n) of noncrossing partitions of a set of n points laying on the boundary of SPostprint (author's final draft
Valuations for matroid polytope subdivisions
We prove that the ranks of the subsets and the activities of the bases of a
matroid define valuations for the subdivisions of a matroid polytope into
smaller matroid polytopes.Comment: 19 pages. 2 figures; added section 6 + other correction
Single-Strip Triangulation of Manifolds with Arbitrary Topology
Triangle strips have been widely used for efficient rendering. It is
NP-complete to test whether a given triangulated model can be represented as a
single triangle strip, so many heuristics have been proposed to partition
models into few long strips. In this paper, we present a new algorithm for
creating a single triangle loop or strip from a triangulated model. Our method
applies a dual graph matching algorithm to partition the mesh into cycles, and
then merges pairs of cycles by splitting adjacent triangles when necessary. New
vertices are introduced at midpoints of edges and the new triangles thus formed
are coplanar with their parent triangles, hence the visual fidelity of the
geometry is not changed. We prove that the increase in the number of triangles
due to this splitting is 50% in the worst case, however for all models we
tested the increase was less than 2%. We also prove tight bounds on the number
of triangles needed for a single-strip representation of a model with holes on
its boundary. Our strips can be used not only for efficient rendering, but also
for other applications including the generation of space filling curves on a
manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200
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