8,558 research outputs found
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
Multilevel refinable triangular PSP-splines (Tri-PSPS)
A multi-level spline technique known as partial shape preserving splines (PSPS) (Li and Tian, 2011) has recently been developed for the design of piecewise polynomial freeform geometric surfaces, where the basis functions of the PSPS can be directly built from an arbitrary set of polygons that partitions a giving parametric domain. This paper addresses a special type of PSPS, the triangular PSPS (Tri-PSPS), where all spline basis functions are constructed from a set of triangles. Compared with other triangular spline techniques, Tri-PSPS have several distinctive features. Firstly, for each given triangle, the corresponding spline basis function for any required degree of smoothness can be expressed in closed-form and directly written out in full explicitly as piecewise bivariate polynomials. Secondly, Tri-PSPS are an additive triangular spline technique, where the spline function built from a given triangle can be replaced with a set of refined spline functions built on a set of smaller triangles that partition the initial given triangle. In addition, Tri-PSPS are a multilevel spline technique, Tri-PSPS surfaces can be designed to have a continuously varying levels of detail, achieved simply by specifying a proper value for the smoothing parameter introduced in the spline functions. In terms of practical implementation, Tri-PSPS are a parallel computing friendly spline scheme, which can be easily implemented on modern programmable GPUs or on high performance computer clusters, since each of the basis functions of Tri-PSPS can be directly computed independent of each other in parallel
Combinatorial cohomology of the space of long knots
The motivation of this work is to define cohomology classes in the space of
knots that are both easy to find and to evaluate, by reducing the problem to
simple linear algebra. We achieve this goal by defining a combinatorial graded
cochain complex, such that the elements of an explicit submodule in the
cohomology define algebraic intersections with some "geometrically simple"
strata in the space of knots. Such strata are endowed with explicit
co-orientations, that are canonical in some sense. The combinatorial tools
involved are natural generalisations (degeneracies) of usual methods using
arrow diagrams.Comment: 20p. 9 fig
Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra
We prove weighted anisotropic analytic estimates for solutions of second
order elliptic boundary value problems in polyhedra. The weighted analytic
classes which we use are the same as those introduced by Guo in 1993 in view of
establishing exponential convergence for hp finite element methods in
polyhedra. We first give a simple proof of the known weighted analytic
regularity in a polygon, relying on a new formulation of elliptic a priori
estimates in smooth domains with analytic control of derivatives. The technique
is based on dyadic partitions near the corners. This technique can successfully
be extended to polyhedra, providing isotropic analytic regularity. This is not
optimal, because it does not take advantage of the full regularity along the
edges. We combine it with a nested open set technique to obtain the desired
three-dimensional anisotropic analytic regularity result. Our proofs are global
and do not require the analysis of singular functions.Comment: 54 page
- …