A C 1 Globally Interpolatory Spline of Arbitrary Topology

He, Ying; Jin, Miao; Gu, Xianfeng; Qin, Hong

doi:10.1007/11567646_25

Cited by 13 publications

(8 citation statements)

References 20 publications

(28 reference statements)

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“…Thus, the evaluation algorithms and other computational procedures are both efficient and robust. They have also showed that any planar spline schemes (defined over an open planar domain) which satisfy the parametric affine invariant property can be straightforwardly extended to manifolds of arbitrary topology within the manifold spline framework [He et al 2006a;He et al 2005;He et al 2006b]. …”

Section: Introductionmentioning

confidence: 99%

Polycube splines

Wang

He²,

et al. 2007

Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling

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Stony Brook (a) Conformal polycube map (b) Polycube T-spline (c) T-junctions on polycube spline (d) Close-up of control pointsFigure 1: Polycube spline for the Isidore Horse model. (a) The conformal polycube map serving as the parametric domain. (b) and (c) Polycube T-splines obtained via affine structure induced by the polycube map. Note that our polycube spline is globally defined as a "onepiece" shape representation without any cutting and gluing work except at the finite number of extraordinary points (corners of the polycube). The extraordinary points are colored in yellow in (b) and (c). The red curves on the spline surface (see (c)) highlight the T-junctions. (d) Close-up of the spline model overlaid with the control points. The polycube T-spline contains 12158 control points. The original model contains 150K vertices. The root-mean-square error is 0.07% of the diagonal of the model. AbstractThis paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric domain. Our rationale for defining spline surfaces over polycubes is that polycubes have rectangular structures everywhere over their domains except a very small number of corner points. The boundary of polycubes can be naturally decomposed into a set of regular structures, which facilitate tensor-product surface definition, GPU-centric geometric computing, and image-based geometric processing. We develop algorithms to construct polycube maps, and show that the introduced polycube map naturally induces the affine structure with a finite number of extraordinary points. Besides its intrinsic rectangular structure, the polycube map may approximate any original scanned data-set with a very low geometric distortion, so our method for building polycube splines is both natural and necessary, as its parametric domain can mimic the geometry of modeled objects in a topologically correct and geometrically meaningful manner. We design a new data structure that facilitates the intuitive and rapid construction of polycube splines in this paper. We demonstrate the polycube splines with applications in surface reconstruction and shape computing.

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Section: Introductionmentioning

confidence: 99%

Polycube splines

Wang

He²,

et al. 2007

Proceedings of the 2007 ACM Symposium on Solid and Physical Modeling

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“…The geometric modeling literature on constructing 𝐺 1 /𝐶 1 surfaces is vast, but due to constraints of our problem, only a few constructions are relevant in our context; [He et al 2005;Powell and Sabin 1977] which we rely on, and [Dahmen 1989] being most closely related. We provide more details in Section 5.…”

Section: Related Workmentioning

confidence: 99%

Algebraic Smooth Occluding Contours

Capouellez

Jiacheng

Hertzmann

et al. 2023

Special Interest Group on Computer Graphics and Interactive Techniques Conference Conference Proceedings

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Input mesh with parametrization Parametric domain layoutContours, cusps and intersections Final contours Projectively transformed mesh Piecewise-Quadratic Surface Figure 1: Our method takes a triangle mesh, and renders smooth occluding contours for a given camera viewpoint. For each view, the algorithm produces a piecewise-quadratic surface, for which we show the occluding contours can be computed algebraically under orthographic projection. We use a projective transformation of the input mesh to handle perspective projection. By precomputing a conformal parameterization and a mapping from the viewpoint to the quadratic coefficients, contours and visibility can be computed very efficiently, in contrast to previous methods that used expensive heuristics or computations to produce smooth contours with accurate visibility.

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“…(If the interior h of the union of the finite elements is not equal to , then the functions in S h vanish on \ h . Details of construction of C 1 splines on arbitrary topology are in [15,16]). Throughout the paper, for a nonnegative integer k, the standard norm in the Sobolev space H k ( ) is denoted by · k , L 2 ( ) = H 0 ( ); H 1 0 ( ) denotes the space of all functions φ ∈ H 1 ( ) with φ = 0 on ∂ ; C denotes a generic positive constant which may depend on r, but which is independent of h, the time-discretization parameter τ and the exact solution of the partial differential equation.…”

Section: Introductionmentioning

confidence: 99%

“…However, for our nonlinear model problem (1.1) in non-divergence form with diffusion coefficient depending linearly or nonlinearly on the unknown solution, the choice of H 2 trial space is natural. Construction of C 1 splines (and hence H 2 trial spaces) is no longer considered to be difficult even on arbitrary topology, in view of the recent work [15,16] and references therein. In comparison with C 0 finite element Galerkin methods, C 1 smoothness of the H 1 -Galerkin approximate solutions leads to significantly smaller linear systems.…”

Section: Introductionmentioning

confidence: 99%

A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

Ganesh

Mustapha

2007

Numer Algor

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We propose and analyze a fully discrete H 1 -Galerkin method with quadrature for nonlinear parabolic advection-diffusion-reaction equations that requires only linear algebraic solvers. Our scheme applied to the special case heat equation is a fully discrete quadrature version of the least-squares method. We prove second order convergence in time and optimal H 1 convergence in space for the computer implementable method. The results of numerical computations demonstrate optimal order convergence of scheme in H k for k = 0, 1, 2.

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A C 1 Globally Interpolatory Spline of Arbitrary Topology

Cited by 13 publications

References 20 publications

Polycube splines

Polycube splines

Algebraic Smooth Occluding Contours

A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

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