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Slides - Computer Graphics and Multimedia
Variational Tangent Plane Intersection for Planar Polygonal Meshing Henrik Zimmer, Marcel Campen, Ralf Herkrath, Leif Kobbelt 1 Computer Graphics Group Henrik Zimmer Motivation 2 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006 Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006 Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007 3 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006 Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006 Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007 4 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006 Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006 Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007 5 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures Supporting + Covering Layer Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006 Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006 Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007 6 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures Supporting + Covering Layer Node/edge simplicity Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006 Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006 Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007 7 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures Supporting + Covering Layer Node/edge simplicity Planar panels Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006 Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006 Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007 8 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures Supporting + Covering Layer Node/edge simplicity Planar panels Discrete Surfaces for Architectural Design. Pottmann, Brell-Cokcan, Wallner. 2006 Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006 Geometry of multi-layer freeform structures for architecture. Pottmann, Liu, Wallner, Bobenko, Wang. 2007 9 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures 10 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures Primal Layer 11 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures Primal Layer Dual Layer 12 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures 13 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) 14 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) 15 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) • Variational TPl 16 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) • Variational TPl – Polygon Mesh Planarization 17 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) • Variational TPl – Dual Support Structures 18 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) • Variational TPl – Variational Shape Approximation 19 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) • Variational TPl Planarization, Dual Support Structures, Shape Approximation 20 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) • Variational TPl Planarization, Dual Support Structures, Shape Approximation 21 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) • Variational TPl Planarization, Dual Support Structures, Shape Approximation 22 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) • Variational TPl Planarization, Dual Support Structures, Shape Approximation 23 Computer Graphics Group Henrik Zimmer Motivation • Multi-Layer Support Structures • Multi-Layer Dual Structures • Tangent Plane Intersection (TPI) • Variational TPl Planarization, Dual Support Structures, Shape Approximation 24 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • 3 planes necessary for unique intersection point 25 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • 3 planes necessary for unique intersection point 26 ∩ = ∩ = ∩ = Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • 3 planes necessary for unique intersection point ∩ = ∩ = ∩ = [Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008] 27 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • 3 planes necessary for unique intersection point 𝒏1 𝒏2 𝒗1 𝒗2 𝒏0 𝒗0 𝒏𝑇0 𝒏𝑇0 𝒗0 𝒏1𝑇 𝒙 = 𝒏1𝑇 𝒗1 𝒏𝑇2 𝒏𝑇2 𝒗2 𝑁𝒙 = 𝒃 Intersection point 𝒙 obtained by inversion 𝒙 = 𝑁 −1 𝒃 [Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008] 28 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • 3 planes necessary for unique intersection point – Positive Curvature: OK [Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008] 29 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • 3 planes necessary for unique intersection point – Low Gaussian Curvature: UNSTABLE [Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008] 30 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • 3 planes necessary for unique intersection point – Low Gaussian Curvature: UNSTABLE Variational Tangent Plane Intersection [Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008] 31 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • 3 planes necessary for unique intersection point – Intersections are predetermined: No design DoFs [Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008] 32 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • 3 planes necessary for unique intersection point – Intersections are predetermined: No design DoFs Variational Tangent Plane Intersection [Planar Hexagonal Meshes by Tangent Plane Intersection, Troche 2008] 33 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 1: Instability, co-planar tangent planes 34 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 1: Instability, co-planar tangent planes 35 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 1: Instability, co-planar tangent planes 36 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 1: Instability, co-planar tangent planes – Intersection point 𝒙 = 𝑁 −1 𝒃 is not well-defined 37 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 1: Instability, co-planar tangent planes – Intersection point 𝒙 = 𝑁 −1 𝒃 is not well-defined – However, 𝑁𝒙 = 𝒃 still holds for all points 38 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 1: Instability, co-planar tangent planes – Intersection point 𝒙 = 𝑁 −1 𝒃 is not well-defined – However, 𝑁𝒙 = 𝒃 still holds for all points from all points we would like to choose our favorite 39 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 2: bad (predetermined) intersections – Intersection is well-defined but position unwanted 40 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 2: bad (predetermined) intersections – Intersection is well-defined but position unwanted – Could obtain more degrees of freedom by: • rotating tangent planes 41 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 2: bad (predetermined) intersections – Intersection is well-defined but position unwanted – Could obtain more degrees of freedom by: • rotating tangent planes 42 Computer Graphics Group Henrik Zimmer Tangent Plane Intersection • Problem 2: bad (predetermined) intersections – Intersection is well-defined but position unwanted – Could obtain more degrees of freedom by: • rotating tangent planes • offsetting tangent planes 43 Computer Graphics Group Henrik Zimmer Variational Tangent Plane Intersection 44 Computer Graphics Group Henrik Zimmer Variational Tangent Plane Intersection • Let 𝑀 = (𝑉, 𝐹) be a triangle mesh 45 Computer Graphics Group Henrik Zimmer Variational Tangent Plane Intersection • Let 𝑀 = (𝑉, 𝐹) be a triangle mesh • Formulate TPI as a constrained optimization: minimize 𝐸 s.t. 𝐶int : 𝑁𝑓 𝒙𝑓 = 𝒃𝑓 ∀𝑓 ∈ 𝐹 where 𝒙𝑓 are the unknown intersection points 46 Computer Graphics Group Henrik Zimmer Variational Tangent Plane Intersection • Let 𝑀 = (𝑉, 𝐹) be a triangle mesh • Formulate TPI as a constrained optimization: minimize 𝐸 s.t. 𝐶int : 𝑁𝑓 𝒙𝑓 = 𝒃𝑓 ∀𝑓 ∈ 𝐹 where 𝒙𝑓 are the unknown intersection points • In non-degenerate configurations the solution is equivalent to the explicit TPI approach 47 Computer Graphics Group Henrik Zimmer Variational Tangent Plane Intersection 𝐸 49 s.t. Computer Graphics Group Henrik Zimmer 𝐶int : 𝑁𝑓 𝒙𝑓 = 𝒃𝑓 ∀𝑓 ∈ 𝐹 Variational Tangent Plane Intersection • Solution to Problem 1: instability – Specify energy with preferred intersection points 𝒑𝑓 𝐸≔ 𝒙𝑓 − 𝒑𝑓 2 𝑓∈𝐹 50 Computer Graphics Group Henrik Zimmer s.t. 𝐶int : 𝑁𝑓 𝒙𝑓 = 𝒃𝑓 ∀𝑓 ∈ 𝐹 Variational Tangent Plane Intersection • Solution to Problem 1: instability – Specify energy with preferred intersection points 𝒑𝑓 • Solution to Problem 2: “bad” intersection points – Introduce variable offsets ℎ𝑣 𝐸≔ 𝒙𝑓 − 𝒑𝑓 2 𝑓∈𝐹 51 Computer Graphics Group Henrik Zimmer s.t. 𝐶int : 𝑁𝑓 𝒙𝑓 = 𝒃𝑓 − 𝒉𝑓 ∀𝑓 ∈ 𝐹 Variational Tangent Plane Intersection • Solution to Problem 1: instability – Specify energy with preferred intersection points 𝒑𝑓 • Solution to Problem 2: “bad” intersection points – Introduce variable offsets ℎ𝑣 and normals 𝒏𝑣 𝐸≔ 𝒙𝑓 − 𝒑𝑓 2 𝑓∈𝐹 52 Computer Graphics Group Henrik Zimmer s.t. 𝐶int : 𝑁𝑓 𝒙𝑓 = 𝒃𝑓 − 𝒉𝑓 ∀𝑓 ∈ 𝐹 𝐶norm : 𝒏𝑣 2 =1 ∀𝑣 ∈ 𝑉 Variational Tangent Plane Intersection • Solution to Problem 1: instability – Specify energy with preferred intersection points 𝒑𝑓 • Solution to Problem 2: “bad” intersection points – Introduce variable offsets ℎ𝑣 and normals 𝒏𝑣 𝐸≔ 𝒙𝑓 − 𝒑𝑓 2 𝑓∈𝐹 s.t. 𝐶int : 𝑁𝑓 𝒙𝑓 = 𝒃𝑓 − 𝒉𝑓 ∀𝑓 ∈ 𝐹 𝐶norm : 𝒏𝑣 2 =1 • So far VTPI is defined for triangle meshes … 53 Computer Graphics Group Henrik Zimmer ∀𝑣 ∈ 𝑉 Multiple Tangent Plane Intersection 54 Computer Graphics Group Henrik Zimmer Multiple Tangent Plane Intersection • 3 planes necessary for unique intersection point 𝒏1 𝒏2 𝒗2 𝒗1 𝒏0 𝒗0 55 Computer Graphics Group Henrik Zimmer 𝒏𝑇0 𝒏𝑇0 𝒗0 𝒏1𝑇 𝒙 = 𝒏1𝑇 𝒗1 𝒏𝑇2 𝒏𝑇2 𝒗2 𝑁𝒙 = 𝒃 Multiple Tangent Plane Intersection • 3 planes necessary for unique intersection point 𝒏1 𝒏2 𝒗2 𝒗1 𝒏3 𝒗3 56 𝒏0 𝒗0 Computer Graphics Group Henrik Zimmer 𝒏𝑇0 𝒏𝑇0 𝒗0 𝒏1𝑇 𝒙 = 𝒏1𝑇 𝒗1 𝒏𝑇2 𝒏𝑇2 𝒗2 𝑁𝒙 = 𝒃 Multiple Tangent Plane Intersection • 3 planes necessary for unique intersection point 𝒏1 𝒏2 𝒗2 𝒗1 𝒏3 𝒗3 𝒏0 𝒗0 𝒏𝑇0 𝒏1𝑇 𝑇 𝒙= 𝒏2 𝒏𝑇3 𝒏𝑇0 𝒗0 𝒏1𝑇 𝒗1 𝒏𝑇2 𝒗2 𝒏𝑇3 𝒗3 • No longer limited to triangle meshes 57 Computer Graphics Group Henrik Zimmer 𝑁𝒙 = 𝒃 Multiple Tangent Plane Intersection • 3 planes necessary for unique intersection point 𝒏1 𝒏2 𝒗2 𝒗1 𝒏3 𝒗3 𝒏0 𝒗0 𝒏𝑇0 𝒏1𝑇 𝒏𝑇2 𝒙 = 𝒏𝑇3 ⋮ 𝒏𝑇0 𝒗0 𝒏1𝑇 𝒗1 𝒏𝑇2 𝒗2 𝒏𝑇3 𝒗3 ⋮ • Works on general polygon meshes 58 Computer Graphics Group Henrik Zimmer 𝑁𝒙 = 𝒃 VTPI for Polygon Mesh Planarization 59 Computer Graphics Group Henrik Zimmer VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) 60 Computer Graphics Group Henrik Zimmer VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) • dual(𝑀) ↦ planar 𝑀 61 Computer Graphics Group Henrik Zimmer VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) • dual(𝑀) ↦ planar 𝑀 FERTILITY 62 Computer Graphics Group Henrik Zimmer VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) • dual(𝑀) ↦ planar 𝑀 FERTILITY 63 [Variational Tangent Plane Intersection] Computer Graphics Group Henrik Zimmer VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) • dual(𝑀) ↦ planar 𝑀 FERTILITY 64 [Variational Tangent Plane Intersection] Computer Graphics Group Henrik Zimmer [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) • dual(𝑀) ↦ planar 𝑀 • Consider quad strips undergoing twists FERTILITY 65 [Variational Tangent Plane Intersection] Computer Graphics Group Henrik Zimmer [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) • dual(𝑀) ↦ planar 𝑀 • Consider quad strips undergoing twists FERTILITY 66 [Variational Tangent Plane Intersection] Computer Graphics Group Henrik Zimmer [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) • dual(𝑀) ↦ planar 𝑀 FELINE 67 [Variational Tangent Plane Intersection] Computer Graphics Group Henrik Zimmer [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) • dual(𝑀) ↦ planar 𝑀 • Consider quads not aligned to princ. curvatures FELINE 68 [Variational Tangent Plane Intersection] Computer Graphics Group Henrik Zimmer [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] VTPI for Polygon Mesh Planarization • 𝑀 ↦ planar dual(𝑀) • dual(𝑀) ↦ planar 𝑀 • Consider quads not aligned to princ. curvatures FELINE 69 [Variational Tangent Plane Intersection] Computer Graphics Group Henrik Zimmer [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] VTPI for Polygon Mesh Planarization • Comparison to other planarization techniques – Optimization (perturbation)-based methods [Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008] [Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006] – Planarizing flow [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] Property Method 70 VTPI Opt. Flow Parameters - - + Extensions + + - Normals + - - Computer Graphics Group Henrik Zimmer VTPI for Polygon Mesh Planarization • Comparison to other planarization techniques – Optimization (perturbation)-based methods [Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008] [Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006] – Planarizing flow [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] Property Method 71 VTPI Opt. Flow Parameters - - + Extensions + + - Normals + - - Computer Graphics Group Henrik Zimmer VTPI for Polygon Mesh Planarization • Comparison to other planarization techniques – Optimization (perturbation)-based methods [Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008] [Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006] – Planarizing flow [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] Property Method 72 VTPI Opt. Flow Parameters - - + Extensions + + - Normals + - - Computer Graphics Group Henrik Zimmer VTPI for Polygon Mesh Planarization • Comparison to other planarization techniques – Optimization (perturbation)-based methods [Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008] [Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006] – Planarizing flow [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] Property Method 73 VTPI Opt. Flow Parameters - - + Extensions + + - Normals + - - Computer Graphics Group Henrik Zimmer VTPI for Polygon Mesh Planarization • Comparison to other planarization techniques – Optimization (perturbation)-based methods [Hexagonal Meshes with Planar Faces. Wang, Liu, Yan, Chan, Ling, Sun. 2008] [Geometric Modeling with Conical Meshes and Developable Surfaces. Liu, Pottmann, Wallner, Yang, Wang. 2006] – Planarizing flow [Discrete Laplacians on general polygonal meshes. Alexa, Wardetzky. 2011] Property Method VTPI Opt. Flow Parameters - - + Extensions + + - Normals + - - • use normals → intersection-free dual structures 74 Computer Graphics Group Henrik Zimmer VTPI for Multi-Layer Dual Structures Input VTPI + preferred offsets + intersection hard-constraints 75 Computer Graphics Group Henrik Zimmer Dual VTPI + preferred offsets TPI VTPI Avoiding (local) Intersections Problem 76 Computer Graphics Group Henrik Zimmer Solution (constraints) Avoiding (local) Intersections Vertex Intersection Problem 77 Solution (constraints) dual primal Computer Graphics Group Henrik Zimmer Avoiding (local) Intersections Vertex Intersection Problem 78 Solution (constraints) dual ℎ>0 primal Computer Graphics Group Henrik Zimmer Avoiding (local) Intersections Solution (constraints) dual ℎ>0 primal Face Intersection Vertex Intersection Problem 79 Computer Graphics Group Henrik Zimmer Avoiding (local) Intersections Solution (constraints) dual ℎ>0 primal 𝒗 Face Intersection Vertex Intersection Problem 80 𝒙 Computer Graphics Group Henrik Zimmer 𝒎 𝒎𝑇 𝒙 − 𝒗 > 0 Avoiding (local) Intersections Solution (constraints) dual ℎ>0 primal 𝒗 Face Intersection Vertex Intersection Problem Edge Intersection 𝒙 81 Computer Graphics Group Henrik Zimmer 𝒎 𝒎𝑇 𝒙 − 𝒗 > 0 Avoiding (local) Intersections Solution (constraints) dual ℎ>0 primal 𝒗 Face Intersection Vertex Intersection Problem 𝒎𝑇 𝒙 − 𝒗 > 0 𝒙 𝒒 Edge Intersection 82 𝒎 Computer Graphics Group Henrik Zimmer 𝒆 𝒒𝑇𝒆 < 0 Results of Multi-Layer Dual Structures 83 Computer Graphics Group Henrik Zimmer T R A I N S TAT I O N 84 Computer Graphics Group Henrik Zimmer Courtesy of www.evolute.at Results of Multi-Layer Dual Structures Results of Multi-Layer Dual Structures Back view ALPINEHUT Side view 85 Computer Graphics Group Henrik Zimmer Conclusion • What is VTPI? 86 Computer Graphics Group Henrik Zimmer Conclusion • What is VTPI? – A variational formulation of tangent plane intersection • Guided intersections of several planes • Useful for geometric problems (e.g. mesh planarization) • Solved by global optimization (freely available solvers) 87 Computer Graphics Group Henrik Zimmer Conclusion • What is VTPI? – A variational formulation of tangent plane intersection • Guided intersections of several planes • Useful for geometric problems (e.g. mesh planarization) • Solved by global optimization (freely available solvers) • What is VTPI not? – A “fix” to topological issues involved in planar meshing • Degeneracies will occur where necessary, e.g. concave (or degenerate) hexagons in hyperbolic surface regions • Energies can sometimes be used to shift such effects … 88 Computer Graphics Group Henrik Zimmer Limitations & Discussion • Output depends on input tessellation and energy 89 Computer Graphics Group Henrik Zimmer Limitations & Discussion • Output depends on input tessellation and energy – Different tessellations, same topology 90 Computer Graphics Group Henrik Zimmer Limitations & Discussion • Output depends on input tessellation and energy – Different tessellations, same topology, same functional 91 Computer Graphics Group Henrik Zimmer Limitations & Discussion • Output depends on input tessellation and energy – Different tessellations, same topology, same functional 92 Computer Graphics Group Henrik Zimmer Limitations & Discussion • Output depends on input tessellation and energy – energies can partly shift some effects on the mesh 93 Computer Graphics Group Henrik Zimmer Limitations & Discussion • Output depends on input tessellation and energy – Same tessellation, same topology, different functional Normal Smoothness 94 Computer Graphics Group Henrik Zimmer Element Fairing Limitations & Discussion • Output depends on input tessellation and energy – Same tessellation, same topology, different functional Normal Smoothness 95 Computer Graphics Group Henrik Zimmer Element Fairing The End Thank you for your attention! 96 Computer Graphics Group Henrik Zimmer
