1 Introduction to Topological Shape Modeling Part I Overview: What is topology?

2 What is Topology? Pliable geometry?! Identifies shapes if they are equivalent under smooth deformation Deformation without object splitting and merging

3 What can Topology do? Roughly classify a variety of shapes Works as a upper layer in hierarchical representation of shapes Classification based on the number of torus holes

4 Upper layer What can Topology do? 1 hole2 holesno hole3 holes Close surfaces …

5 Examples Connectivity Graphs Shape structure Decomposition into Cells Shape embedding in space Knots and links

6 Connectivity Isomorphism between graphs All graphs are isomorphic. Complete graph

7 Shape Structure Decomposing a shape into topological entities Topological structure of a torus Vertex Edge Face peak pass pit Morse theory

8 Embedding in Space Objects have restrictions in space. Different between unknotted and knotted circles

9 How does the topology classify shapes? Prepare special equivalence relations Geometry: equal(=) Topology: ??? Find quotient space based on the equivalence relation

10 Grouping Numbers If we use equal(=) for grouping … Too detailed to understand the global distribution

11 Grouping Numbers If we classify into even and odd … (If we compare remainders when the nubmers is divided by 2.) Even numbers: The remainder is 0 when divided by 2 Odd numbers: The remainder is 1 when divided by 2 Only two groups!!

12 Grouping Numbers If we compare remainders when the numbers are divided by The remainder is 0 when divided by 3 The remainder is 1 When divided by 3 The remainder is 2 When divided by 3

13 Grouping Shapes What is an equivalence relation for shapes? Equivelent?Equivalent? Topology provides good equivalence relations for rough shape classification. Equivalent?

14 Grouping Shapes Equivalent if they can change into each other without splitting and merging

15 Grouping Shapes Answer is as follows:

16 What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more …

17 What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more …

18 Topological Surface Design peak pass pit Upper layer in hierarchical representation

19 Topological Surface Design peak pass pit Upper layer in hierarchical representation

Solid Modeling Topological Surface Design

Solid Modeling Examples Torus

Solid Modeling Examples: Toy dogLetters

Solid Modeling Examples Double-layered swirl

24 What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more …

Eurographics Terrain Surface Analysis Rendered images Mt. FujiLake Ashi

Eurographics Terrain Surface Analysis Peaks, passes, pits, and contours Mt. FujiLake Ashi

Eurographics Terrain Surface Analysis Ridge and ravine lines Mt. FujiLake Ashi

Eurographics Terrain Surface Analysis Surface networks Mt. FujiLake Ashi

Eurographics Terrain Surface Analysis Reeb graphs (Contour trees) Mt. FujiLake Ashi

Eurographics Terrain Surface Analysis Reeb graphs (Contour trees) Mt. FujiLake Ashi

31 Wireframe representation Surface Analysis Topological skeleton (Reeb graph)

32 Surface Analysis Reeb graphs (Topological skeletons)

33 Reeb graph (Topological skeleton) Surface Analysis

34 What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more …

35 Tracing Isosurface Transitions Topological volume skeleton Splitting and merging of isosurfaces Volume skeleton tree (VST)

36 Volume Analysis Topological analysis of volume Transfer function Design Based on topological analysis

37 Embedding-dependent Rendering TF by default VST-based Embedding- dependent Visualizing complicated inner structure

38 What is topology applied to? Surface design Surface analysis Volume analysis Morphing design and more …

Computer and Graphics From a human head to a tiger head Morphing = Surface + Time

Computers and Graphics From a bunny to a cat Morphing = Surface + Time

Pacific Graphics Topological Evolution? Need to specify the topology in evolution!!

Pacific Graphics “ 8 ” - “ 0 ” - “ V ” - “ 11 ” - “ H ” - “ B ” - “ A ” Topological Curve Morphing

Pacific Graphics Topological Surface Morphing From two spheres to one sphere

Pacific Graphics Morphing design From torus to sphere by cutting

Pacific Graphics The opening to a void within a solid is closed. Results