Refinable Polycube G-Splines
|
|
|
- Denis Fletcher
- 6 years ago
- Views:
Transcription
1 Refinable Polycube G-Splines Martin Sarov Jörg Peters University of Florida NSF CCF M. Sarov, J. Peters (UF) SMI / 21
2 Background Quest Surfaces with rational linear reparameterization = as close as possible to regular splines M. Sarov, J. Peters (UF) SMI / 21
3 Geometric Continuity Geometric Continuity G 1 = C 1 continuity after change of variables M. Sarov, J. Peters (UF) SMI / 21
4 Geometric Continuity Geometric Continuity G 1 = C 1 continuity after change of variables Simplest non-affine change of variables: rational linear q = p(ρ) M. Sarov, J. Peters (UF) SMI / 21
5 Geometric Continuity Geometric Continuity G 1 = C 1 continuity after change of variables Simplest non-affine change of variables: rational linear q = p(ρ) Implies linear scaling by ω: p(t, 0) = q(0, t), ω(t) 1 p (t, 0) = 2 p (t, 0) + 1 q (0, t), ( ω(t) := (1 t)ω 0 + tω 1, ω 0, ω 1 R, M. Sarov, J. Peters (UF) SMI / 21
6 Geometric Continuity Geometric Continuity G 1 = C 1 continuity after change of variables Simplest non-affine change of variables: rational linear q = p(ρ) Implies linear scaling by ω: p(t, 0) = q(0, t), ω(t) 1 p (t, 0) = 2 p (t, 0) + 1 q (0, t), ( ω(t) := (1 t)ω 0 + tω 1, ω 0, ω 1 R, : rational linear = closest to regular spline (ω = const) : maximizes the number of control handles M. Sarov, J. Peters (UF) SMI / 21
7 Geometric Continuity Geometric Continuity G 1 = C 1 continuity after change of variables Simplest non-affine change of variables: rational linear q = p(ρ) Implies linear scaling by ω: p(t, 0) = q(0, t), ω(t) 1 p (t, 0) = 2 p (t, 0) + 1 q (0, t), ( ω(t) := (1 t)ω 0 + tω 1, ω 0, ω 1 R, : rational linear = closest to regular spline (ω = const) : maximizes the number of control handles : ω 0 = ω 1 or p(0, 0) and p(1, 0) have the same valence M. Sarov, J. Peters (UF) SMI / 21
8 Geometric Continuity Geometric Continuity G 1 = C 1 continuity after change of variables Simplest non-affine change of variables: rational linear q = p(ρ) Implies linear scaling by ω: p(t, 0) = q(0, t), ω(t) 1 p (t, 0) = 2 p (t, 0) + 1 q (0, t), ( ω(t) := (1 t)ω 0 + tω 1, ω 0, ω 1 R, : rational linear = closest to regular spline (ω = const) : maximizes the number of control handles : ω 0 = ω 1 or p(0, 0) and p(1, 0) have the same valence [PS15]: valence restricted to 3, 6 M. Sarov, J. Peters (UF) SMI / 21
9 Polycube connectivity and control handles Outline 1 Polycube connectivity and control handles 2 Surface Construction 3 Multi-resolution M. Sarov, J. Peters (UF) SMI / 21
10 Polycube connectivity and control handles Restricted Connectivity: Polycube Configurations n {3, 4, 5, 6}: M. Sarov, J. Peters (UF) SMI / 21
11 Polycube connectivity and control handles 2x2 split: Control Handles c i M. Sarov, J. Peters (UF) SMI / 21
12 Polycube connectivity and control handles 2x2 split: Control Handles c i Control handles c i Boundary points b i (Bézier) M. Sarov, J. Peters (UF) SMI / 21
13 Polycube connectivity and control handles 2x2 split: Control Handles c i Edge Recovery: c i b i M. Sarov, J. Peters (UF) SMI / 21
14 Polycube connectivity and control handles 2x2 split: Control Handles c i Edge Recovery: c i b i smooth? M. Sarov, J. Peters (UF) SMI / 21
15 Polycube connectivity and control handles 2x2 split: Control Handles c i M. Sarov, J. Peters (UF) SMI / 21
16 Surface Construction Outline 1 Polycube connectivity and control handles 2 Surface Construction 3 Multi-resolution M. Sarov, J. Peters (UF) SMI / 21
17 Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients (technical). M. Sarov, J. Peters (UF) SMI / 21
18 Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients Apparent valence (mimic 3 6). 2c p b α,11 10 = b α+1, b α 1,11 10, c p := cos 2π p. M. Sarov, J. Peters (UF) SMI / 21
19 Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients Apparent valence Assign labels: n, m edge. M. Sarov, J. Peters (UF) SMI / 21
20 Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients Apparent valence Assign labels: n, m edge Simple G 1 constraints work for n, m {3, 4, 5, 6} except n, 4, 4, m with n m. M. Sarov, J. Peters (UF) SMI / 21
21 Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients Apparent valence Assign labels: n, m edge Simple G 1 constraints work for n, m {3, 4, 5, 6} except n, 4, 4, m with n m. Construction steps:. M. Sarov, J. Peters (UF) SMI / 21
22 Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients Apparent valence Assign labels: n, m edge Simple G 1 constraints work for n, m {3, 4, 5, 6} except n, 4, 4, m with n m. M. Sarov, J. Peters (UF) SMI / 21
23 Surface Construction Smooth PGS surfaces M. Sarov, J. Peters (UF) SMI / 21
24 Surface Construction Smooth PGS surfaces M. Sarov, J. Peters (UF) SMI / 21
25 Multi-resolution Outline 1 Polycube connectivity and control handles 2 Surface Construction 3 Multi-resolution M. Sarov, J. Peters (UF) SMI / 21
26 Multi-resolution Multi-resolution of Polycube G1 -splines M. Sarov, J. Peters (UF) SMI / 21
27 Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? M. Sarov, J. Peters (UF) SMI / 21
28 Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces M. Sarov, J. Peters (UF) SMI / 21
29 Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces : Catmull-Clark on polycube quad net M. Sarov, J. Peters (UF) SMI / 21
30 Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces : Catmull-Clark on polycube quad net : C 1 bicubic spline refinement: depends on b i M. Sarov, J. Peters (UF) SMI / 21
31 Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces : Catmull-Clark on polycube quad net : C 1 bicubic spline refinement: depends on b i : if c i is set by PGS (= satisfies G1) then c i is unchanged by Edge Recovery + PGS. M. Sarov, J. Peters (UF) SMI / 21
32 Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces : Catmull-Clark on polycube quad net : C 1 bicubic spline refinement: depends on b i : if c i is set by PGS (= satisfies G1) then c i is unchanged by Edge Recovery + PGS. de Casteljau preserves G1 of finer c i M. Sarov, J. Peters (UF) SMI / 21
33 Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces : Catmull-Clark on polycube quad net : C 1 bicubic spline refinement: depends on b i : if c i is set by PGS (= satisfies G1) then c i is unchanged by Edge Recovery + PGS. de Casteljau preserves G1 of finer c i Edge Recovery + PGS = same surface M. Sarov, J. Peters (UF) SMI / 21
34 Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces : Catmull-Clark on polycube quad net : C 1 bicubic spline refinement: depends on b i : if c i is set by PGS (= satisfies G1) then c i is unchanged by Edge Recovery + PGS. de Casteljau preserves G1 of finer c i Edge Recovery + PGS = same surface perturb c i = different smooth surface M. Sarov, J. Peters (UF) SMI / 21
35 Multi-resolution Summary Smooth surface covering irregularities of valence n = 3, 4, 5, 6 in a C 1 bi-3 spline surface M. Sarov, J. Peters (UF) SMI / 21
36 Multi-resolution Summary Smooth surface covering irregularities of valence n = 3, 4, 5, 6 in a C 1 bi-3 spline surface Control Handles = as for C 1 bi-3 spline (larger support) M. Sarov, J. Peters (UF) SMI / 21
37 Multi-resolution Summary Smooth surface covering irregularities of valence n = 3, 4, 5, 6 in a C 1 bi-3 spline surface Control Handles = as for C 1 bi-3 spline (larger support) Refinable reproduces surfaces at finer level QUESTIONS? M. Sarov, J. Peters (UF) SMI / 21
38 Multi-resolution Summary Smooth surface covering irregularities of valence n = 3, 4, 5, 6 in a C 1 bi-3 spline surface Control Handles = as for C 1 bi-3 spline (larger support) Refinable reproduces surfaces at finer level QUESTIONS? M. Sarov, J. Peters (UF) SMI / 21
39 Multi-resolution Multiresolution: algebraic challenge Challenge: global system, : additional constraint ω( k 2 l ) ( bα;k 1, b α;k 1,1 20 2b α;k b α;k,1 20 ) = 0. }{{} C 2 boundary : large support of c i handles : decay fast M. Sarov, J. Peters (UF) SMI / 21
40 Multi-resolution PGSER M. Sarov, J. Peters (UF) SMI / 21
41 Multi-resolution PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients: d d b(u, v) := b ij Bi d (u)bj d (v), i=0 j=0 p 01 + q 10 = ω 0 p 10 + (2 ω 0 )p 00, (1) p 11 + q 11 = ρ(ω 0, ω 1 ), (2) p 21 + q 12 = σ(ω 0, ω 1 ), (3) p 31 + q 13 = (2 + ω 1 )p 30 ω 1 p 20. (4) ρ(ω 0, ω 1 ) := 1 ( ) 2ω0 p 20 ω 1 p 00 + (6 2ω 0 + ω 1 )p 10 3 σ(ω 0, ω 1 ) := 1 ) ω0 p 30 2ω 1 p 10 + (6 ω 0 + 2ω 1 )p 20. 3( M. Sarov, J. Peters (UF) SMI / 21
42 Multi-resolution What goes wrong when n, 4, 4, m with n m? M. Sarov, J. Peters (UF) SMI / 21
Smooth Multi-Sided Blending of bi-2 Splines
Smooth Multi-Sided Blending of bi-2 Splines Kȩstutis Karčiauskas Jörg Peters Vilnius University University of Florida K. Karčiauskas, J. Peters (VU, UF) SMI14: Bi-3/4 Caps for bi-2 Splines 1 / 18 Quad
Refinable C 1 spline elements for irregular quad layout
Refinable C 1 spline elements for irregular quad layout Thien Nguyen Jörg Peters University of Florida NSF CCF-0728797, NIH R01-LM011300 T. Nguyen, J. Peters (UF) GMP 2016 1 / 20 Outline 1 Refinable, smooth,
Refinable Polycube G-Splines
Refinable Polycube G-Splines Martin Sarova and Jo rg Petersa a University of Florida Abstract Polycube G-splines form a -manifold guided by a mesh of quadrilateral faces such that at most six quads meet
Subdivision 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
On 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
Smooth Bi-3 Spline Surfaces with fewest knots
Smooth Bi-3 Spline Surfaces with fewest knots Jianhua Fan, Jörg Peters Department of CISE, University of Florida, USA 3603 Abstract Converting a quadrilateral input mesh into a C 1 surface with one bi-3
Finite curvature continuous polar patchworks
Finite curvature continuous polar patchworks Kȩstutis Karčiauskas 0, Jörg Peters 1 0 Vilnius University, 1 University of Florida Abstract. We present an algorithm for completing a C 2 surface of up to
Curve 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
Subdivision 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
Subdivision 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
Subdivision 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
INF3320 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
Normals 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,
Example: 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
Advanced 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),
Surfaces 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
Subdivision. 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
BICUBIC UNIFORM B-SPLINE SURFACE REFINEMENT
On-Line Geometric Modeling Notes BICUBIC UNIFORM B-SPLINE SURFACE REFINEMENT Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science Uniersity of California, Dais Oeriew
Non-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
Smooth Surfaces from 4-sided Facets
Smooth Surfaces from -sided Facets T. L. Ni, Y. Yeo, A. Myles, V. Goel and J. Peters Abstract We present a fast algorithm for converting quad meshes on the GPU to smooth surfaces. Meshes with 1,000 input
Subdivision Depth Computation for Extra-Ordinary Catmull-Clark Subdivision Surface Patches
Subdivision Depth Computation for Extra-Ordinary Catmull-Clark Subdivision Surface Patches Fuhua Frank Cheng,GangChen, and Jun-Hai Yong University of Kentucky, Lexington, KY, USA Tsinghua University, Beijing,
Normals 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
Curvature of Approximating Curve Subdivision Schemes
Curvature of Approximating Curve Subdivision Schemes Kȩstutis Karčiauskas 1 and Jörg Peters 2 1 Vilnius University, 2 University of Florida Abstract. The promise of modeling by subdivision is to have simple
REAL-TIME SMOOTH SURFACE CONSTRUCTION ON THE GRAPHICS PROCESSING UNIT
REAL-TIME SMOOTH SURFACE CONSTRUCTION ON THE GRAPHICS PROCESSING UNIT By TIANYUN NI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
09 - 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
Approximating 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
Improved shape for refinable surfaces with singularly parameterized irregularities
Improved shape for refinable surfaces with singularly parameterized irregularities Kȩstutis Karčiauskas a and Jörg Peters b a Naugarduko 24, Vilnius University, LT-2006, Vilnius, Lithuania b Dept CISE,
G 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
Polar 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
Smooth 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
Curves, Surfaces and Recursive Subdivision
Department of Computer Sciences Graphics Fall 25 (Lecture ) Curves, Surfaces and Recursive Subdivision Conics: Curves and Quadrics: Surfaces Implicit form arametric form Rational Bézier Forms Recursive
Subdivision 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
Curves 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
Pairs of Bi-Cubic Surface Constructions Supporting Polar Connectivity
Pairs of Bi-Cubic Surface Constructions Supporting Polar Connectivity Ashish Myles a, Kestutis Karčiauskas b Jörg Peters a a Department of CISE, University of Florida b Department of Mathematics and Informatics,
Recursive 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.
Volume 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
Using 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
CS354 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
Hierarchical Grid Conversion
Hierarchical Grid Conversion Ali Mahdavi-Amiri, Erika Harrison, Faramarz Samavati Abstract Hierarchical grids appear in various applications in computer graphics such as subdivision and multiresolution
UNIVERSITY 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
Subdivision 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,
GL9: Engineering Communications. GL9: CAD techniques. Curves Surfaces Solids Techniques
436-105 Engineering Communications GL9:1 GL9: CAD techniques Curves Surfaces Solids Techniques Parametric curves GL9:2 x = a 1 + b 1 u + c 1 u 2 + d 1 u 3 + y = a 2 + b 2 u + c 2 u 2 + d 2 u 3 + z = a
2001, Denis Zorin. Subdivision Surfaces
200, Denis Zorin Subdivision Surfaces 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
Know it. Control points. B Spline surfaces. Implicit surfaces
Know it 15 B Spline Cur 14 13 12 11 Parametric curves Catmull clark subdivision Parametric surfaces Interpolating curves 10 9 8 7 6 5 4 3 2 Control points B Spline surfaces Implicit surfaces Bezier surfaces
Bézier and B-spline volumes Project of Dissertation
Department of Algebra, Geometry and Didactics of Mathematics Faculty of Mathemathics, Physics and Informatics Comenius University, Bratislava Bézier and B-spline volumes Project of Dissertation Martin
Trimming 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
Approximating Subdivision Surfaces with Gregory Patches for Hardware Tessellation
Approximating Subdivision Surfaces with Gregory Patches for Hardware Tessellation Charles Loop Microsoft Research Scott Schaefer Texas A&M University Tianyun Ni NVIDIA Ignacio Castaño NVIDIA Goal Real-Time
Refinable C 1 spline elements for irregular quad layout
Refinable C 1 spline elements for irregular quad layout Thien Nguyen a, Jörg Peters a, a Dept CISE, University of Florida, USA Abstract Building on a result of U. Reif on removable singularities, we construct
Curved PN Triangles. Alex Vlachos Jörg Peters
1 Curved PN Triangles Alex Vlachos AVlachos@ati.com Jörg Peters jorg@cise.ufl.edu Outline 2 Motivation Constraints Surface Properties Performance Demo Quick Demo 3 Constraints 4 Software Developers Must
Spline 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
Non-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
Computer 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
From 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
Subdivision 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,
EFFICIENT, TIGHT BOUNDING VOLUMES FOR SUBDIVISION SURFACES
EFFICIENT, TIGHT BOUNDING VOLUMES FOR SUBDIVISION SURFACES By XIAOBIN WU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
Computer Graphics Lecture 5
1 / 25 Computer Graphics Lecture 5 Dr. Marc Eduard Frîncu West University of Timisoara Mar 27th 2012 2 / 25 Outline 1 Graphical Primitives Drawing Surfaces 2 Assignment 3 Recap 3 / 25 Graphical Primitives
Computer Graphics Spline and Surfaces
Computer Graphics 2016 10. Spline and Surfaces Hongxin Zhang State Key Lab of CAD&CG, Zhejiang University 2016-12-05 Outline! Introduction! Bézier curve and surface! NURBS curve and surface! subdivision
Subdivision 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
A Heuristic Offsetting Scheme for Catmull-Clark Subdivision Surfaces
1 A Heuristic Offsetting Scheme for Catmull-Clark Subdivision Surfaces Jianzhong Wang 1, Fuhua Cheng 1,2 1 University of Kentucky, jwangf@uky.edu, cheng@cs.uky.edu 2 ational Tsing Hua University, cheng@cs.uky.edu
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
Gauss curvature. curvature lines, needle plot. vertices of progenitor solid are unit size. Construction Principle. edge. edge. vertex. Patch.
The Platonic Spheroids Jorg Peters and Leif Kobbelt Abstract. We present a gallery of simple curvature continuous surfaces that possess the topological structure of the Platonic solids. These spherelike
Semi-Regular 4 8 Refinement and Box Spline Surfaces
Semi-Regular 4 8 Refinement and Box Spline Surfaces LUIZ VELHO IMPA Instituto de Matemática Pura e Aplicada, lvelho@visgraf.impa.br Abstract. In this paper we introduce a new mesh refinement method for
08 - Designing Approximating Curves
08 - Designing Approximating Curves Acknowledgement: Olga Sorkine-Hornung, Alexander Sorkine-Hornung, Ilya Baran Last time Interpolating curves Monomials Lagrange Hermite Different control types Polynomials
Digital Geometry Processing. Computer Graphics CMU /15-662
Digital Geometry Processing Computer Graphics CMU 15-462/15-662 Last time: Meshes & Manifolds Mathematical description of geometry - simplifying assumption: manifold - for polygon meshes: fans, not fins
Refinable bivariate quartic and quintic C 2 -splines for quadrilateral subdivisions
Refinable bivariate quartic and quintic C 2 -splines for quadrilateral subdivisions Charles K. Chui, Qingtang Jiang Department of Mathematics and Computer Science University of Missouri St. Louis St. Louis,
Efficient GPU Rendering of Subdivision Surfaces. Tim Foley,
Efficient GPU Rendering of Subdivision Surfaces Tim Foley, 2017-03-02 Collaborators Activision Wade Brainerd Stanford Matthias Nießner NVIDIA Manuel Kraemer Henry Moreton 2 Subdivision surfaces are a powerful
3 Multiresolution by Local Least Squares: The Diagrammatic Approach
Multiresolution by Local Least Squares: The Diagrammatic Approach Richard Bartels Ali Mahdavi-Amiri Faramarz Samavati October 9, 205 Abstract In [2,, 20, 2] the authors explored a construction to produce
2D Spline Curves. CS 4620 Lecture 18
2D Spline Curves CS 4620 Lecture 18 2014 Steve Marschner 1 Motivation: smoothness In many applications we need smooth shapes that is, without discontinuities So far we can make things with corners (lines,
Geometry Processing & Geometric Queries. Computer Graphics CMU /15-662
Geometry Processing & Geometric Queries Computer Graphics CMU 15-462/15-662 Last time: Meshes & Manifolds Mathematical description of geometry - simplifying assumption: manifold - for polygon meshes: fans,
DESIGNING AND IMPLEMENTING A SURFACE MODELING SYSTEM
DESIGNING AND IMPLEMENTING A SURFACE MODELING SYSTEM By LE-JENG SHIUE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
3D 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
CS130 : Computer Graphics Curves (cont.) Tamar Shinar Computer Science & Engineering UC Riverside
CS130 : Computer Graphics Curves (cont.) Tamar Shinar Computer Science & Engineering UC Riverside Blending Functions Blending functions are more convenient basis than monomial basis canonical form (monomial
Figure 5.1: Spline and ducks.
Chapter 5 B-SPLINE CURVES Most shapes are simply too complicated to define using a single Bézier curve. A spline curve is a sequence of curve segments that are connected together to form a single continuous
Guided spline surfaces
Computer Aided Geometric Design 26 (2009) 105 116 www.elsevier.com/locate/cagd Guided spline surfaces K. Karčiauskas a, J. Peters b, a Department of Mathematics and Informatics, Naugarduko 24, 03225 Vilnius,
An Interpolatory Subdivision for Volumetric Models over Simplicial Complexes
An Interpolatory Subdivision for Volumetric Models over Simplicial Complexes Yu-Sung Chang Kevin T. McDonnell Hong Qin Department of Computer Science State University of New York at Stony Brook {yusung
Modeling. Simulating the Everyday World
Modeling Simulating the Everyday World Three broad areas: Modeling (Geometric) = Shape Animation = Motion/Behavior Rendering = Appearance Page 1 Geometric Modeling 1. How to represent 3d shapes Polygonal
Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches
Approximating Catmull-Clark Subdivision Surfaces with Bicubic Patches CHARLES LOOP Microsoft Research and SCOTT SCHAEFER Texas A&M University We present a simple and computationally efficient algorithm
SEMIREGULAR 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
The Essentials of CAGD
The Essentials of CAGD Chapter 6: Bézier Patches Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/essentials-cagd c 2 Farin & Hansford The
Curves D.A. Forsyth, with slides from John Hart
Curves D.A. Forsyth, with slides from John Hart Central issues in modelling Construct families of curves, surfaces and volumes that can represent common objects usefully; are easy to interact with; interaction
Computer Graphics Fundamentals. Jon Macey
Computer Graphics Fundamentals Jon Macey jmacey@bournemouth.ac.uk http://nccastaff.bournemouth.ac.uk/jmacey/ 1 1 What is CG Fundamentals Looking at how Images (and Animations) are actually produced in
Curves and Surfaces Computer Graphics I Lecture 9
15-462 Computer Graphics I Lecture 9 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] February 19, 2002 Frank Pfenning Carnegie
An 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
Evaluation 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
Curves & Surfaces. Last Time? Progressive Meshes. Selective Refinement. Adjacency Data Structures. Mesh Simplification. Mesh Simplification
Last Time? Adjacency Data Structures Curves & Surfaces Geometric & topologic information Dynamic allocation Efficiency of access Mesh Simplification edge collapse/vertex split geomorphs progressive transmission
CAGD PACKAGE FOR MATHEMATICA AND ITS USAGE IN THE TEACHING
5. KONFERENCE O GEOMETRII A POČÍTAČOVÉ GRAFICE Bohumír Bastl CAGD PACKAGE FOR MATHEMATICA AND ITS USAGE IN THE TEACHING Abstract This talk presents a new package for Wolfram s Mathematica which provides
Fast Parallel Construction of Smooth Surfaces from Meshes with Tri/Quad/Pent Facets
Eurographics Symposium on Geometry Processing 2008 Pierre Alliez and Szymon Rusinkiewicz (Guest Editors Volume 27 (2008, Number 5 Fast Parallel Construction of Smooth Surfaces from Meshes with Tri/Quad/Pent
Mesh Representations & Geometry Processing
Lecture 10/11: Mesh Representations & Geometry Processing Computer Graphics and Imaging UC Berkeley A Small Triangle Mesh 8 vertices, 12 triangles A Large Triangle Mesh David Digital Michelangelo Project
BS-Patch: Constrained Bezier Parametric Patch
BS-Patch: Constrained Bezier Parametric Patch VACLAV SKALA, VIT ONDRACKA Department of Computer Science and Engineering University of West Bohemia Univerzitni 8, CZ 06 14 Plzen CZECH REPUBLIC skala@kiv.zcu.cz
The 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
C 2 SplinesCoveringPolarConfigurations
C 2 SplinesCoveringPolarConfigurations Ashish Myles a a CS Dept, New York University, 75 Broadway, 2th Floor, New York, NY 0003, USA Jörg Peters b b CISE Dept, Bldg 42, Rm E30, University of Florida, Gainesville,
CS-184: Computer Graphics
CS-184: Computer Graphics Lecture #12: Curves and Surfaces Prof. James O Brien University of California, Berkeley V2007-F-12-1.0 Today General curve and surface representations Splines and other polynomial
Surface and Solid Geometry. 3D Polygons
Surface and Solid Geometry D olygons Once we know our plane equation: Ax + By + Cz + D = 0, we still need to manage the truncation which leads to the polygon itself Functionally, we will need to do this
A 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
Subdivision 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
On the Linear Independence of Truncated Hierarchical Generating Systems
On the Linear Independence of Truncated Hierarchical Generating Systems Urška Zore a, Bert Jüttler a,, Jiří Kosinka b a Institute of Applied Geometry, Johannes Kepler University, Linz, Austria b Johann
Shape 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
Subdivision 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)!
Lecture 3 Mesh. Dr. Shuang LIANG. School of Software Engineering Tongji University Spring 2013
Lecture 3 Mesh Dr. Shuang LIANG School of Software Engineering Tongji University Spring 2013 Today s Topics Overview Mesh Acquisition Mesh Data Structures Subdivision Surfaces Today s Topics Overview Mesh
Combining 4- and 3-direction Subdivision
Combining - and 3-direction Subdivision Jörg Peters and Le-Jeng Shiue University of Florida -3 direction subdivision combines quad and triangle meshes. On quad submeshes it applies a -direction alternative