u 0+u 2 new boundary vertex
|
|
- Phebe Thornton
- 5 years ago
- Views:
Transcription
1 Combined Subdivision Schemes for the design of surfaces satisfying boundary conditions Adi Levin School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. Abstract We present a new kind of subdivision schemes, which we call combined subdivision schemes. Such schemes are used to rene a given control net, taking into consideration boundary conditions on the limit surface. Several schemes are presented that generate limit surfaces interpolating given boundary curves and cross boundary derivatives. The interpolated boundary curves are arbitrary parametric curves, and are not restricted to subdivision curves or to spline curves. Key words: conditions. Subdivision schemes, Combined subdivision schemes, boundary Introduction The attractivity of bi-variate subdivision schemes for the representation of smooth surfaces of three-dimensional objects, comes from their simplicity and adaptivity to meshes of general topology. This is their main advantage over the commonly used tensor-product surfaces. Subdivision schemes operate as follows: Given a set of initial control points P 0, the scheme performs local operations to calculate a ner set of points P. This process is repeated recursively, generating P n with n = 2; 3; :::. In the limit, the points P n converge to a limit surface s. In this setting, the way to force boundary conditions on the limit surface, is to translate these conditions to conditions on P 0. Subdivision schemes have already been used by Schweitzer [6], by Nasri [5] and by Kobbelt [2] to dene 'open surfaces', i.e. surfaces with boundaries. In [5], the boundaries of limit surfaces of Doo-Sabin's subdivision scheme, are identied as quadratic B-spline curves, under certain conditions on the control polygon. Preprint submitted to Elsevier Preprint 23 November 998
2 In [6], Loop's scheme is extended to a non-uniform subdivision scheme whose limit surfaces can have creases and corner points, by introducing dierent subdivision rules near special areas of the control net. In [2] the tensor product four point scheme is extended to an interpolatory subdivision scheme that works for quadrilateral meshes of arbitrary topology. All of these papers deal with specics schemes, and remain within the classical subdivision setting, providing a rule that renes control points by local operations on them. In this paper, we present a new kind of subdivision schemes, named combined subdivision schemes. A combined scheme calculates the points on the next re- nement level, taking into consideration given boundary conditions. In this setting, conditions on the limit surface can be translated to conditions on the scheme itself, rather than on the initial control points as done in [5,6,2] reviewed above. A key requirement is that combined subdivision schemes remain as local and as simple as ordinary subdivision schemes. As a result, the process of constraining the boundaries of the limit surface becomes considerably simpler. The combined schemes that we design dier from ordinary subdivision schemes only in their action near the boundary of the control net. Away from the boundary we apply known subdivision schemes. The input to a combined subdivision scheme consists of a set of initial control points P 0 and boundary conditions c on the limit surface. The scheme performs local operations on P 0 and on c to calculate a ner set of points P. This process is repeated recursively, generating P n with n = 2; 3; :::. This paper demonstrates the implementation of this idea in several cases. For each example, we constructed special schemes that operate near the boundary and we checked their smoothness using the theory developed in [3]. 2 Interpolating smooth boundary curves with Catmull Clark's scheme Catmull Clark's scheme is dened in [] over meshes with no boundaries, as follows: Every 'face' of the mesh is divided into 4-sided 'faces', as shown in gure. There are three types of vertices in the resulting mesh (see gure 2): (a) A vertex that corresponds to an old 'face' is calculated by averaging the control points dening the 'face'. (b) A vertex that corresponds to an old edge is calculated by averaging the control points corresponding to the two old vertices at the ends of 2
3 Fig.. Quadrilateral subdivision made by Catmull Clark's scheme (a) (b) Wn (c) Fig. 2. Stencils of Catmull Clark's scheme (up to division by the sum of the weights) that edge, and to the two new vertices that correspond to the two old 'faces' near that edge. (c) A vertex corresponding to an old vertex is calculated by averaging the control points corresponding to the old vertex, the old beighboring vertices and the new vertices corresponding to the 'faces' sharing the old vertex. The old vertex is given a higher weight w n, which depends on the valency n of that vertex. The topology of the new mesh can be deduced from gure. The value w 4 = 8 guarantees that the scheme generates cubic tensor product splines, except near extraordinary vertices (i.e. vertices with valency 6= 4). There are dierent possible formulas for w n when n 6= 4. In the following examples, we take w n = n(n? 2) for n 6= 4. For the combined scheme we use a simple rule called boundary sampling: We assume that the boundary condition is given in the form of a smooth parametric curve c(u) and that every boundary vertex lies on the given curve. A new 3
4 c(u 0 ) old boundary edge c(u ) C c( u 0+u ) 2 new boundary vertex Fig. 3. Boundary sampling Fig. 4. The input to the combined scheme boundary vertex is calculated by appropriate sampling of the given curve, as depicted in gure 3. The combined scheme is dened as follows: Apply Catmull Clark's scheme everywhere except for the boundary of the mesh. On the boundary of the mesh apply Boundary sampling. Figure 4 shows an example for the input to the combined scheme: A curve, with an initial control net. The boundary vertices lie on the given curve. Figure 5 shows the resulting meshes after four iterations of subdivision. In the next example, we use the combined scheme to design a surface with three prescribed boundaries. Figure 6 shows the input to the combined scheme, along with two iterations of the scheme, and the surface obtained after 4 iterations. A sucient condition for C 2 smoothness of the limit surface near the boundary is that (a) The given boundary curves are C 2 (b) The boundary vertices are placed on the boundary curves uniformly with respect to the curves' parametrization (c) Every boundary vertex has valency 3. The required analysis tools can be found in [3]. 4
5 Fig. 5. Four iterations of the combined scheme Fig. 6. A surface with three prescribed boundaries 3 Interpolating piecewise smooth boundary curves with Loop's scheme Loop's scheme is dened in [4] over triangulations with no boundaries, as follows: Every face of the triangulation is divided into four triangles, as shown in gure 7. There are two types of vertices in the resulting mesh: 5
6 Fig. 7. Triangular subdivision made by Loop's scheme Wn 3 3 Fig. 8. The stencils of Loop's scheme () A vertex that corresponds to an old edge. (2) A vertex that corresponds to an old vertex. The two stencils that are used to calculate the new vertices are depicted in gure 8. The weight W n depends on the valency n of the vertex, through the formula W n = 8n 5? (3 + 2 cos(? n: () 2 ))2 8 n We extend this scheme to a combined scheme by the rule of boundary sampling depicted in gure 3. Near corners (breakpoints) of the boundary curves, we introduce the a dierent stencil, for a new vertex which corresponds to an old edge that starts with a corner vertex, as depicted in gure 9. The choice of this scheme guarantees that the limit surfaces are almost G 2 near corners of the boundary (An application of the Loop's stencil given in gure 8 near the breakpoints generates limit surfaces with innite curvature on the boundary breakpoints). In the following example, we start with a boundary curve that has ve breakpoints. The initial mesh contains the ve breakpoints, and another internal vertex. Figure 0 shows a top view of the subdivision process. Figure shows a side view of two subdivision iterations, and the surface obtained after 4 iterations. We have shown that the limit surface is C 2 near the boundary, except at the 6
7 Fig. 9. The stencil used near corner vertices Fig. 0. Three iterations of the combined scheme corners of the boundary curves, where we only have a necessary condition for smoothness. 4 Prescribing cross-boundary derivatives with Catmull Clark's scheme The schemes we have developed in sections 2 and 3 accept as boundary conditions only given boundary curves. In this section, we introduce a scheme that accepts as input boundary conditions of higher order. Our motivation is to prescribe not only the boundary curves, but also the normal vectors (or the tangent plane) of the surface at the boundary, in order to force tangency to a given neighboring surface. Assuming that the neighboring surface is a parametric surface where the 7
8 Fig.. Two iterations of the combined scheme, and the limit surface boundary curve is an iso-parametric curve, we have the boundary curve given as a parametric curve, and the surface partial derivatives on the boundary (with respect to the second parameter) given as a second function. We call the values of this function cross boundary derivatives. The following scheme is based on the scheme developed in section 2, namely - Catmull Clark's scheme away from the boundary, and boundary sampling on the boundary. Here we add two more stencils for the calculation of one layer of new vertices near the boundary. These vertices should depend on given cross-boundary derivatives. There are two kinds of such vertices: (a) Vertices that correspond to old edges with a vertex on the boundary. (b) Vertices that correspond to old boundary faces. The stencils for the two cases are depicted in gure 2. The bold lines signify boundary edges. The arrow signies the given cross boundary derivative. The weight 6 in stencil (b) multiplies the value of the boundary curve at the averaged parameter. The cross boundary derivative is multiplied by W L and added to the average of the control points with unnormalized weights depicted in gure 2. The weight W L depends on the level L of subdivision, and is given by W L = C 2?L : (2) The constant C aects the shape of the resulting surface. The cross boundary derivative has to be given as a function of the correspond- 8
9 (a) (b) W L W L 6 Fig. 2. Stencils that consider cross-boundary derivatives Fig. 3. An initial mesh for a blend surface between three cylinders ing parameter on the boundary curve. A sample of that function between two existing vertices is taken at the averaged parameter value. In the following example, we use this combined scheme to create a blending surface between given cylinders. Figure 3 shows the initial mesh, with the three given cylinders that dene the cross boundary derivatives. Figure 4 shows three iterations of subdivision, and the limit surface. In a similar way, we can blend any number of cylinders. Figures 5 and 6 show the initial mesh and the limit surface, Blending 5 and 7 cylinders respectively. 5 Conclusions Combined subdivision schemes can be used to generate surfaces satisfying different kinds of boundary conditions. These schemes oer a simple alternative to other methods for prescribing boundary conditions, since they do not involve the solution of systems of equations, but only explicit evaluation of the boundary conditions. 9
10 Fig. 4. Four iterations of the combined scheme, and the limit surface Fig. 5. Blending ve cylinders Fig. 6. Blending seven cylinders Their main advantage over existing methods is that they enable the representation of surfaces that satsify arbitrary boundary conditions, while in existing surface representations the boundary curves are restricted to subdivision curves or spline curves. 0
11 Acknowledgement This work is sponsored by the Israeli Ministry of Science. I also wish to thank my supervisor in this work, Nira Dyn, for her guidance and helpful comments. References [] D. Doo and M. Sabin. Behaviour of recursive division surface near extraordinary points. Computer Aided Design, 0:356{360, 978. [2] L. Kobbelt, T. Hesse, H. Prautzsch, and K. Schweizerhof. Interpolatory subdivsion on open quadrilateral nets with arbitrary topology. Computer Graphics Forum, 5:409{420, 996. Eurographics '96 issue. [3] A. Levin. Analysis of combined subdivision schemes for the design of surfaces satisfying boundary conditions [4] C. Loop. Smooth spline surfaces based on triangles. Master's thesis, University of Utah, Department of Mathematics, 987. [5] A. H. Nasri. Free-form curve generation by recursive subdivision of polygonal complexes. presented at Nashville, November 997. [6] J. Schweitzer. Analysis and Applications of Subdivision Surfaces. PhD thesis, University of Washington, Seattle, 996.
Interpolatory 3-Subdivision
EUROGRAPHICS 2000 / M. Gross and F.R.A. Hopgood (Guest Editors) Volume 19 (2000), Number 3 Interpolatory 3-Subdivision U. Labsik G. Greiner Computer Graphics Group University of Erlangen-Nuremberg Am Weichselgarten
More informationSubdivision Curves and Surfaces: An Introduction
Subdivision Curves and Surfaces: An Introduction Corner Cutting De Casteljau s and de Boor s algorithms all use corner-cutting procedures. Corner cutting can be local or non-local. A cut is local if it
More informationSubdivision Curves and Surfaces
Subdivision Surfaces or How to Generate a Smooth Mesh?? Subdivision Curves and Surfaces Subdivision given polyline(2d)/mesh(3d) recursively modify & add vertices to achieve smooth curve/surface Each iteration
More informationUsing Semi-Regular 4 8 Meshes for Subdivision Surfaces
Using Semi-Regular 8 Meshes for Subdivision Surfaces Luiz Velho IMPA Instituto de Matemática Pura e Aplicada Abstract. Semi-regular 8 meshes are refinable triangulated quadrangulations. They provide a
More informationSubdivision Surfaces
Subdivision Surfaces 1 Geometric Modeling Sometimes need more than polygon meshes Smooth surfaces Traditional geometric modeling used NURBS Non uniform rational B-Spline Demo 2 Problems with NURBS A single
More informationExample: Loop Scheme. Example: Loop Scheme. What makes a good scheme? recursive application leads to a smooth surface.
Example: Loop Scheme What makes a good scheme? recursive application leads to a smooth surface 200, Denis Zorin Example: Loop Scheme Refinement rule 200, Denis Zorin Example: Loop Scheme Two geometric
More informationSubdivision Surfaces
Subdivision Surfaces 1 Geometric Modeling Sometimes need more than polygon meshes Smooth surfaces Traditional geometric modeling used NURBS Non uniform rational B-Spline Demo 2 Problems with NURBS A single
More informationRecursive Subdivision Surfaces for Geometric Modeling
Recursive Subdivision Surfaces for Geometric Modeling Weiyin Ma City University of Hong Kong, Dept. of Manufacturing Engineering & Engineering Management Ahmad Nasri American University of Beirut, Dept.
More informationINF3320 Computer Graphics and Discrete Geometry
INF3320 Computer Graphics and Discrete Geometry More smooth Curves and Surfaces Christopher Dyken, Michael Floater and Martin Reimers 10.11.2010 Page 1 More smooth Curves and Surfaces Akenine-Möller, Haines
More informationUNIVERSITY OF CALGARY. Subdivision Surfaces. Advanced Geometric Modeling Faramarz Samavati
Subdivision Surfaces Surfaces Having arbitrary Topologies Tensor Product Surfaces Non Tensor Surfaces We can t find u-curves and v-curves in general surfaces General Subdivision Coarse mesh Subdivision
More informationG 2 Bezier Crust on Quad Subdivision Surfaces
Pacific Graphics (2013) B. Levy, X. Tong, and K. Yin (Editors) Short Papers G 2 Bezier Crust on Quad Subdivision Surfaces paper 1348 Figure 1: Two examples of Bezier crust applied on Catmull-Clark subdivision
More informationCurves and Surfaces 2
Curves and Surfaces 2 Computer Graphics Lecture 17 Taku Komura Today More about Bezier and Bsplines de Casteljau s algorithm BSpline : General form de Boor s algorithm Knot insertion NURBS Subdivision
More informationTechnical Report. Removing polar rendering artifacts in subdivision surfaces. Ursula H. Augsdörfer, Neil A. Dodgson, Malcolm A. Sabin.
Technical Report UCAM-CL-TR-689 ISSN 1476-2986 Number 689 Computer Laboratory Removing polar rendering artifacts in subdivision surfaces Ursula H. Augsdörfer, Neil A. Dodgson, Malcolm A. Sabin June 2007
More informationJoe Warren, Scott Schaefer Rice University
Joe Warren, Scott Schaefer Rice University Polygons are a ubiquitous modeling primitive in computer graphics. Their popularity is such that special purpose graphics hardware designed to render polygons
More informationSurfaces for CAGD. FSP Tutorial. FSP-Seminar, Graz, November
Surfaces for CAGD FSP Tutorial FSP-Seminar, Graz, November 2005 1 Tensor Product Surfaces Given: two curve schemes (Bézier curves or B splines): I: x(u) = m i=0 F i(u)b i, u [a, b], II: x(v) = n j=0 G
More informationSmooth Patching of Refined Triangulations
Smooth Patching of Refined Triangulations Jörg Peters July, 200 Abstract This paper presents a simple algorithm for associating a smooth, low degree polynomial surface with triangulations whose extraordinary
More informationCS354 Computer Graphics Surface Representation IV. Qixing Huang March 7th 2018
CS354 Computer Graphics Surface Representation IV Qixing Huang March 7th 2018 Today s Topic Subdivision surfaces Implicit surface representation Subdivision Surfaces Building complex models We can extend
More informationNormals of subdivision surfaces and their control polyhedra
Computer Aided Geometric Design 24 (27 112 116 www.elsevier.com/locate/cagd Normals of subdivision surfaces and their control polyhedra I. Ginkel a,j.peters b,,g.umlauf a a University of Kaiserslautern,
More informationCurve Corner Cutting
Subdivision ision Techniqueses Spring 2010 1 Curve Corner Cutting Take two points on different edges of a polygon and join them with a line segment. Then, use this line segment to replace all vertices
More informationlinearize discretize Galerkin optimize sample
Fairing by Finite Dierence Methods Leif Kobbelt Abstract. We propose an ecient and exible scheme to fairly interpolate or approximate the vertices of a given triangular mesh. Instead of generating a piecewise
More informationNormals of subdivision surfaces and their control polyhedra
Normals of subdivision surfaces and their control polyhedra I. Ginkel, a, J. Peters b, and G. Umlauf a, a University of Kaiserslautern, Germany b University of Florida, Gainesville, FL, USA Abstract For
More informationVolume Enclosed by Example Subdivision Surfaces
Volume Enclosed by Example Subdivision Surfaces by Jan Hakenberg - May 5th, this document is available at vixra.org and hakenberg.de Abstract Simple meshes such as the cube, tetrahedron, and tripod frequently
More informationTernary Butterfly Subdivision
Ternary Butterfly Subdivision Ruotian Ling a,b Xiaonan Luo b Zhongxian Chen b,c a Department of Computer Science, The University of Hong Kong b Computer Application Institute, Sun Yat-sen University c
More informationModified Catmull-Clark Methods for Modelling, Reparameterization and Grid Generation
Modified Catmull-Clark Methods for Modelling, Reparameterization and Grid Generation Karl-Heinz Brakhage RWTH Aachen, 55 Aachen, Deutschland, Email: brakhage@igpm.rwth-aachen.de Abstract In this paper
More informationSubdivision based Interpolation with Shape Control
Subdivision based Interpolation with Shape Control Fengtao Fan University of Kentucky Deparment of Computer Science Lexington, KY 40506, USA ffan2@uky.edu Fuhua (Frank) Cheng University of Kentucky Deparment
More informationSubdivision curves and surfaces. Brian Curless CSE 557 Fall 2015
Subdivision curves and surfaces Brian Curless CSE 557 Fall 2015 1 Reading Recommended: Stollnitz, DeRose, and Salesin. Wavelets for Computer Graphics: Theory and Applications, 1996, section 6.1-6.3, 10.2,
More informationSubdivision surfaces. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Subdivision surfaces University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Reading Recommended: Stollnitz, DeRose, and Salesin. Wavelets for Computer Graphics: Theory and Applications,
More informationSubdivision overview
Subdivision overview CS4620 Lecture 16 2018 Steve Marschner 1 Introduction: corner cutting Piecewise linear curve too jagged for you? Lop off the corners! results in a curve with twice as many corners
More informationFast Rendering of Subdivision Surfaces
Fast Rendering of Subdivision Surfaces Kari Pulli (Univ. of Washington, Seattle, WA) Mark Segal (SGI) Abstract Subdivision surfaces provide a curved surface representation that is useful in a number of
More informationAn Efficient Data Structure for Representing Trilateral/Quadrilateral Subdivision Surfaces
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 3, No 3 Sofia 203 Print ISSN: 3-9702; Online ISSN: 34-408 DOI: 0.2478/cait-203-0023 An Efficient Data Structure for Representing
More informationGeneralizing the C 4 Four-directional Box Spline to Surfaces of Arbitrary Topology Luiz Velho Abstract. In this paper we introduce a new scheme that g
Generalizing the C 4 Four-directional Box Spline to Surfaces of Arbitrary Topology Luiz Velho Abstract. In this paper we introduce a new scheme that generalizes the four-directional box spline of class
More information09 - Designing Surfaces. CSCI-GA Computer Graphics - Fall 16 - Daniele Panozzo
9 - Designing Surfaces Triangular surfaces A surface can be discretized by a collection of points and triangles Each triangle is a subset of a plane Every point on the surface can be expressed as an affine
More informationComputergrafik. Matthias Zwicker Universität Bern Herbst 2016
Computergrafik Matthias Zwicker Universität Bern Herbst 2016 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling 2 Piecewise Bézier curves Each
More informationTaxonomy of interpolation. constraints on recursive subdivision. Ahmad H. Nasri 1, Malcolm A. Sabin 2. 1 Introduction
1 Introduction Taxonomy of interpolation constraints on recursive subdivision surfaces Ahmad H. Nasri 1, Malcolm A. Sabin 2 1 Department of Mathematics and Computer Science, American University of Beirut,
More informationSubdivision on Arbitrary Meshes: Algorithms and Theory
Subdivision on Arbitrary Meshes: Algorithms and Theory Denis Zorin New York University 719 Broadway, 12th floor, New York, USA E-mail: dzorin@mrl.nyu.edu Subdivision surfaces have become a standard geometric
More information3D Modeling Parametric Curves & Surfaces. Shandong University Spring 2013
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2013 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels
More informationAn interpolating 4-point C 2 ternary stationary subdivision scheme
Computer Aided Geometric Design 9 (2002) 8 www.elsevier.com/locate/comaid An interpolating 4-point C 2 ternary stationary subdivision scheme M.F Hassan a,, I.P. Ivrissimitzis a, N.A. Dodgson a,m.a.sabin
More informationDesign by Subdivision
Bridges 2010: Mathematics, Music, Art, Architecture, Culture Design by Subdivision Michael Hansmeyer Department for CAAD - Institute for Technology in Architecture Swiss Federal Institute of Technology
More informationG 2 Interpolation for Polar Surfaces
1 G 2 Interpolation for Polar Surfaces Jianzhong Wang 1, Fuhua Cheng 2,3 1 University of Kentucky, jwangf@uky.edu 2 University of Kentucky, cheng@cs.uky.edu 3 National Tsinhua University ABSTRACT In this
More information3D Modeling Parametric Curves & Surfaces
3D Modeling Parametric Curves & Surfaces Shandong University Spring 2012 3D Object Representations Raw data Point cloud Range image Polygon soup Solids Voxels BSP tree CSG Sweep Surfaces Mesh Subdivision
More informationCover Page. Title: Surface Approximation Using Geometric Hermite Patches Abstract:
Cover Page Title: Surface Approximation Using Geometric Hermite Patches Abstract: A high-order-of-approximation surface patch is used to construct continuous, approximating surfaces. This patch, together
More informationParameterization. Michael S. Floater. November 10, 2011
Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point
More informationThe goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a
The goal is the definition of points with numbers and primitives with equations or functions. The definition of points with numbers requires a coordinate system and then the measuring of the point with
More informationOn Smooth Bicubic Surfaces from Quad Meshes
On Smooth Bicubic Surfaces from Quad Meshes Jianhua Fan and Jörg Peters Dept CISE, University of Florida Abstract. Determining the least m such that one m m bi-cubic macropatch per quadrilateral offers
More informationRemoving Polar Rendering Artifacts in Subdivision Surfaces
This is an electronic version of an article published in Journal of Graphics, GPU, and Game Tools, Volume 14, Issue 2 pp. 61-76, DOI: 10.1080/2151237X.2009.10129278. The Journal of Graphics, GPU, and Game
More informationSubdivision of Curves and Surfaces: An Overview
Subdivision of Curves and Surfaces: An Overview Ben Herbst, Karin M Hunter, Emile Rossouw Applied Mathematics, Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland,
More informationApproximating Catmull-Clark Subdivision Surfaces with Bicubic Patches
Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches Charles Loop Microsoft Research Scott Schaefer Texas A&M University April 24, 2007 Technical Report MSR-TR-2007-44 Microsoft Research
More informationComputergrafik. Matthias Zwicker. Herbst 2010
Computergrafik Matthias Zwicker Universität Bern Herbst 2010 Today Curves NURBS Surfaces Parametric surfaces Bilinear patch Bicubic Bézier patch Advanced surface modeling Piecewise Bézier curves Each segment
More informationNear-Optimum Adaptive Tessellation of General Catmull-Clark Subdivision Surfaces
Near-Optimum Adaptive Tessellation of General Catmull-Clark Subdivision Surfaces Shuhua Lai and Fuhua (Frank) Cheng (University of Kentucky) Graphics & Geometric Modeling Lab, Department of Computer Science,
More informationHoneycomb Subdivision
Honeycomb Subdivision Ergun Akleman and Vinod Srinivasan Visualization Sciences Program, Texas A&M University Abstract In this paper, we introduce a new subdivision scheme which we call honeycomb subdivision.
More informationSpline Surfaces, Subdivision Surfaces
CS-C3100 Computer Graphics Spline Surfaces, Subdivision Surfaces vectorportal.com Trivia Assignment 1 due this Sunday! Feedback on the starter code, difficulty, etc., much appreciated Put in your README
More informationPolar Embedded Catmull-Clark Subdivision Surface
Polar Embedded Catmull-Clark Subdivision Surface Anonymous submission Abstract In this paper, a new subdivision scheme with Polar embedded Catmull-Clark mesh structure is presented. In this new subdivision
More informationLocal Modification of Subdivision Surfaces Based on Curved Mesh
Local Modification of Subdivision Surfaces Based on Curved Mesh Yoshimasa Tokuyama Tokyo Polytechnic University tokuyama@image.t-kougei.ac.jp Kouichi Konno Iwate University konno@cis.iwate-u.ac.jp Junji
More informationCurves and Surfaces for Computer-Aided Geometric Design
Curves and Surfaces for Computer-Aided Geometric Design A Practical Guide Fourth Edition Gerald Farin Department of Computer Science Arizona State University Tempe, Arizona /ACADEMIC PRESS I San Diego
More informationUntil now we have worked with flat entities such as lines and flat polygons. Fit well with graphics hardware Mathematically simple
Curves and surfaces Escaping Flatland Until now we have worked with flat entities such as lines and flat polygons Fit well with graphics hardware Mathematically simple But the world is not composed of
More informationSubdivision. Outline. Key Questions. Subdivision Surfaces. Advanced Computer Graphics (Spring 2013) Video: Geri s Game (outside link)
Advanced Computer Graphics (Spring 03) CS 83, Lecture 7: Subdivision Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs83/sp3 Slides courtesy of Szymon Rusinkiewicz, James O Brien with material from Denis
More informationAdvanced Graphics. Subdivision Surfaces. Alex Benton, University of Cambridge Supported in part by Google UK, Ltd
Advanced Graphics Subdivision Surfaces Alex Benton, University of Cambridge A.Benton@damtp.cam.ac.uk Supported in part by Google UK, Ltd NURBS patches aren t the greatest NURBS patches are nxm, forming
More informationRobustness of Boolean operations on subdivision-surface models
Robustness of Boolean operations on subdivision-surface models Di Jiang 1, Neil Stewart 2 1 Université de Montréal, Dép t. Informatique CP6128, Succ. CentreVille, Montréal, H3C 3J7, Qc, Canada jiangdi@umontreal.ca
More information1. Introduction. 2. Parametrization of General CCSSs. 3. One-Piece through Interpolation. 4. One-Piece through Boolean Operations
Subdivision Surface based One-Piece Representation Shuhua Lai Department of Computer Science, University of Kentucky Outline. Introduction. Parametrization of General CCSSs 3. One-Piece through Interpolation
More informationSmooth Surface Reconstruction using Doo-Sabin Subdivision Surfaces
Smooth Surface Reconstruction using Doo-Sabin Subdivision Surfaces Fuhua (Frank) Cheng, Fengtao Fan, Conglin Huang, Jiaxi Wang Department of Computer Science, University of Kentucky, Lexington, KY 40506,
More informationA Continuous 3-D Medial Shape Model with Branching
A Continuous 3-D Medial Shape Model with Branching Timothy B. Terriberry Guido Gerig Outline Introduction The Generic 3-D Medial Axis Review of Subdivision Surfaces Boundary Reconstruction Edge Curves
More informationPhysically-Based Modeling and Animation. University of Missouri at Columbia
Overview of Geometric Modeling Overview 3D Shape Primitives: Points Vertices. Curves Lines, polylines, curves. Surfaces Triangle meshes, splines, subdivision surfaces, implicit surfaces, particles. Solids
More informationSEMIREGULAR PENTAGONAL SUBDIVISIONS
SEMIREGULAR PENTAGONAL SUBDIVISIONS ERGUN AKLEMAN & VINOD SRINIVASAN Visualization Sciences Program Texas A&M University ZEKI MELEK & PAUL EDMUNDSON Computer Science Department Abstract Triangular and
More informationAdvanced Modeling 2. Katja Bühler, Andrej Varchola, Eduard Gröller. March 24, x(t) z(t)
Advanced Modeling 2 Katja Bühler, Andrej Varchola, Eduard Gröller March 24, 2014 1 Parametric Representations A parametric curve in E 3 is given by x(t) c : c(t) = y(t) ; t I = [a, b] R z(t) where x(t),
More informationNon-Uniform Recursive Doo-Sabin Surfaces
Non-Uniform Recursive Doo-Sabin Surfaces Zhangjin Huang a,b,c,, Guoping Wang d,e a School of Computer Science and Technology, University of Science and Technology of China, PR China b Key Laboratory of
More informationLocal Mesh Operators: Extrusions Revisited
Local Mesh Operators: Extrusions Revisited Eric Landreneau Computer Science Department Abstract Vinod Srinivasan Visualization Sciences Program Texas A&M University Ergun Akleman Visualization Sciences
More informationTaxonomy of interpolation constraints on recursive subdivision curves
1 Introduction Taxonomy of interpolation constraints on recursive subdivision curves Ahmad H. Nasri 1, Malcolm A. Sabin 2 1 Dept. of Math. & Computer Science, American University of Beirut, Beirut, PO
More informationMA 323 Geometric Modelling Course Notes: Day 36 Subdivision Surfaces
MA 323 Geometric Modelling Course Notes: Day 36 Subdivision Surfaces David L. Finn Today, we continue our discussion of subdivision surfaces, by first looking in more detail at the midpoint method and
More informationNon-Uniform Recursive Doo-Sabin Surfaces (NURDSes)
Non-Uniform Recursive Doo-Sabin Surfaces Zhangjin Huang 1 Guoping Wang 2 1 University of Science and Technology of China 2 Peking University, China SIAM Conference on Geometric and Physical Modeling Doo-Sabin
More informationFrom curves to surfaces. Parametric surfaces and solid modeling. Extrusions. Surfaces of revolution. So far have discussed spline curves in 2D
From curves to surfaces Parametric surfaces and solid modeling CS 465 Lecture 12 2007 Doug James & Steve Marschner 1 So far have discussed spline curves in 2D it turns out that this already provides of
More informationPostprocessing of Compressed 3D Graphic Data
Journal of Visual Communication and Image Representation 11, 80 92 (2000) doi:10.1006/jvci.1999.0430, available online at http://www.idealibrary.com on Postprocessing of Compressed 3D Graphic Data Ka Man
More information3D Modeling techniques
3D Modeling techniques 0. Reconstruction From real data (not covered) 1. Procedural modeling Automatic modeling of a self-similar objects or scenes 2. Interactive modeling Provide tools to computer artists
More informationconsisting of compact sets. A spline subdivision scheme generates from such
Spline Subdivision Schemes for Compact Sets with Metric Averages Nira Dyn and Elza Farkhi Abstract. To dene spline subdivision schemes for general compact sets, we use the representation of spline subdivision
More informationGrid Generation and Grid Conversion by Subdivision Schemes
Grid Generation and Grid Conversion by Subdivision Schemes Karl Heinz Brakhage Institute for Geometry and Applied Mathematics RWTH Aachen University D-55 Aachen brakhage@igpm.rwth-aachen.de Abstract In
More informationSubdivision surfaces for CAD: integration through parameterization and local correction
Workshop: New trends in subdivision and related applications September 4 7, 212 Department of Mathematics and Applications, University of Milano-Bicocca, Italy Subdivision surfaces for CAD: integration
More informationECE 600, Dr. Farag, Summer 09
ECE 6 Summer29 Course Supplements. Lecture 4 Curves and Surfaces Aly A. Farag University of Louisville Acknowledgements: Help with these slides were provided by Shireen Elhabian A smile is a curve that
More informationShape optimization of smooth surfaces with arbitrary topology
International conference on Innovative Methods in Product Design June 15 th 17 th, 2011, Venice, Italy Shape optimization of smooth surfaces with arbitrary topology Przemysław Kiciak (a) (a) Institut Matematyki
More informationSubdivision Surfaces. Homework 1: Questions on Homework? Last Time? Today. Tensor Product. What s an illegal edge collapse?
Homework 1: Questions/Comments? Subdivision Surfaces Questions on Homework? Last Time? What s an illegal edge collapse? Curves & Surfaces Continuity Definitions 2 3 C0, G1, C1, C 1 a b 4 Interpolation
More informationParameterization of triangular meshes
Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to
More informationInformation Coding / Computer Graphics, ISY, LiTH. Splines
28(69) Splines Originally a drafting tool to create a smooth curve In computer graphics: a curve built from sections, each described by a 2nd or 3rd degree polynomial. Very common in non-real-time graphics,
More informationSubdivision Surfaces. Homework 1: Questions/Comments?
Subdivision Surfaces Homework 1: Questions/Comments? 1 Questions on Homework? What s an illegal edge collapse? 1 2 3 a b 4 7 To be legal, the ring of vertex neighbors must be unique (have no duplicates)!
More informationQUADRATIC UNIFORM B-SPLINE CURVE REFINEMENT
On-Line Geometric Modeling Notes QUADRATIC UNIFORM B-SPLINE CURVE REFINEMENT Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis Overview
More informationSubdivision Surfaces. Homework 1: Last Time? Today. Bilinear Patch. Tensor Product. Spline Surfaces / Patches
Homework 1: Questions/Comments? Subdivision Surfaces Last Time? Curves & Surfaces Continuity Definitions Spline Surfaces / Patches Tensor Product Bilinear Patches Bezier Patches Trimming Curves C0, G1,
More informationEvaluation of Loop Subdivision Surfaces
Evaluation of Loop Subdivision Surfaces Jos Stam Alias wavefront, Inc. 8 Third Ave, 8th Floor, Seattle, WA 980, U.S.A. jstam@aw.sgi.com Abstract This paper describes a technique to evaluate Loop subdivision
More informationApproximate Geodesics on Smooth Surfaces of Arbitrary Topology
Approximate Geodesics on Smooth Surfaces of Arbitrary Topology Paper ID: 418 Category: Technical Paper The 6th International Symposium on Visual Computing (ISCV10) Las Vegas, Nevada, November 29 - December
More informationCS354 Computer Graphics Surface Representation III. Qixing Huang March 5th 2018
CS354 Computer Graphics Surface Representation III Qixing Huang March 5th 2018 Today s Topic Bspline curve operations (Brief) Knot Insertion/Deletion Subdivision (Focus) Subdivision curves Subdivision
More informationTriangle Mesh Subdivision with Bounded Curvature and the Convex Hull Property
Triangle Mesh Subdivision with Bounded Curvature and the Convex Hull Property Charles Loop cloop@microsoft.com February 1, 2001 Technical Report MSR-TR-2001-24 The masks for Loop s triangle subdivision
More informationCurves and Surfaces. Shireen Elhabian and Aly A. Farag University of Louisville
Curves and Surfaces Shireen Elhabian and Aly A. Farag University of Louisville February 21 A smile is a curve that sets everything straight Phyllis Diller (American comedienne and actress, born 1917) Outline
More informationFINAL REPORT. Tessellation, Fairing, Shape Design, and Trimming Techniques for Subdivision Surface based Modeling
FINAL REPORT Tessellation, Fairing, Shape Design, and Trimming Techniques for Subdivision Surface based Modeling (DMS-0422126) PI: Fuhua (Frank) Cheng Department of Computer Science College of Engineering
More informationA subdivision scheme for hexahedral meshes
A subdivision scheme for hexahedral meshes Chandrajit Bajaj Department of Computer Sciences, University of Texas Scott Schaefer Department of Computer Science, Rice University Joe Warren Department of
More informationTo Do. Resources. Algorithm Outline. Simplifications. Advanced Computer Graphics (Spring 2013) Surface Simplification: Goals (Garland)
Advanced omputer Graphics (Spring 213) S 283, Lecture 6: Quadric Error Metrics Ravi Ramamoorthi To Do Assignment 1, Due Feb 22. Should have made some serious progress by end of week This lecture reviews
More informationSubdivision Surfaces
Subdivision Surfaces CS 4620 Lecture 31 Cornell CS4620 Fall 2015 1 Administration A5 due on Friday Dreamworks visiting Thu/Fri Rest of class Surfaces, Animation, Rendering w/ prior instructor Steve Marschner
More informationComputer Graphics Curves and Surfaces. Matthias Teschner
Computer Graphics Curves and Surfaces Matthias Teschner Outline Introduction Polynomial curves Bézier curves Matrix notation Curve subdivision Differential curve properties Piecewise polynomial curves
More informationTrimming for Subdivision Surfaces
Trimming for Subdivision Surfaces Nathan Litke a,1 Adi Levin b,2 Peter Schröder a,3 a Caltech, Pasadena, CA 91125, USA b Tel Aviv University, Tel Aviv 69978, Israel Abstract Trimming is an important primitive
More informationSubdivision Scheme Tuning Around Extraordinary Vertices
Subdivision Scheme Tuning Around Extraordinary Vertices Loïc Barthe Leif Kobbelt Computer Graphics Group, RWTH Aachen Ahornstrasse 55, 52074 Aachen, Germany Abstract In this paper we extend the standard
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More informationMatrix-valued 4-point Spline and 3-point Non-spline Interpolatory Curve Subdivision Schemes
Matrix-valued 4-point Spline and -point Non-spline Interpolatory Curve Subdivision Schemes Charles K. Chui, Qingtang Jiang Department of Mathematics and Computer Science University of Missouri St. Louis
More informationA Study on Subdivision Scheme-Draft. Kwan Pyo Ko Dongseo University Busan, South Korea
A Study on Subdivision Scheme-Draft Kwan Pyo Ko Dongseo University Busan, South Korea April 30, 007 Contents Introduction 9 Subdivision of Univariate Data 3. Definitions and Basic Results.............................
More informationExact Evaluation Of Catmull-Clark Subdivision Surfaces At Arbitrary Parameter Values
Exact Evaluation Of Catmull-Clark Subdivision Surfaces At Arbitrary Parameter Values Jos Stam Alias wavefront Inc Abstract In this paper we disprove the belief widespread within the computer graphics community
More informationIn this course we will need a set of techniques to represent curves and surfaces in 2-d and 3-d. Some reasons for this include
Parametric Curves and Surfaces In this course we will need a set of techniques to represent curves and surfaces in 2-d and 3-d. Some reasons for this include Describing curves in space that objects move
More information