1 Subdivision Depth Computation for Catmull-Clark Subdivision Surfaces Fuhua (Frank) Cheng University of Kentucky, Lexington, KY Junhai Yong Tsinghua University, Beijing, China

2 Outline: 1.Introduction 2.Subdivision depth computation for regular CCSS patches 3.For extra-ordinary CCSS patches 4.Examples 5.Concluding Remarks

3 Introduction: Recent focus Recent focus : Subdivision Surfaces The de-facto standard for generating freeform curves and surfaces of arbitrary topology (in visualization and animation application)

4 Graphical Modeling Games Animation

5 Pixar’s Renderman Alias|Wavefront’s Maya Nichimen’s Mirai Newtek’s Lightwave 3D Used as primary representation scheme in:

6 What is a subdivision surface? Triangular Loop Butterfly Quadrilateral Catmull-Clark Doo-Sabin

7 Repeatedly refining the given control mesh to get M 0,M 1,M 2,M 3, limit surface (subdivision surface) M0M0 M1M1 M2M2 M3M3 S = M ∞

8 What so special? One piece representation™ (arbitrary topology) Multi-resolution (Scalability) Code Simplicit y Covers both polygon form and surface form (Uniformity) Numerical stability

9 Why is One Piece Representation ™ Good? Data Management:Simpler Rendering:More efficient Machining: More precise Animation: Crack free

10 Graphic Modeling Games Animation What is missing? CAD/CAM

11 Why? 1. No parameterization 2. No error control 3. No adaptive tessellation

12 Without error control No CAD/CAM applications Without parameterization Difficult to perform picking, rendering, texture mapping Without adaptive tessellation Too expensive to use

13 What is error control?

14 Error Control: Given ε > 0, when would ║M n - S║< ε ? M0M0 M1M1 M2M2 MnMn S = M ∞ Cross-Sectional View Limit Surface ε

15 Subdivision Depth Computation For Regular CCSS Patches: 2 nd Order Forward Difference B A C

16 Cubic B-spline Curve Segment

17 Bicubic B-spline Surface Patch

18 Bicubic B-spline Surface Patch

19 Recurrence Formula

20 Hence, for We need Subdivision Depth

21 Subdivision Depth Computation For Extra-Ordinary CCSS Patches: Ω - Partition based approach Extra-ordinary point (0,0) 1 1 v u

22 Subdivision of a Extra-ordinary patch

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24 Two Regions: –Vicinity of Extra-Ordinary point –The remaining (regular) regions (0,0) 1 1 v u

25

26 For vicinity of Extra-Ordinary point:

27 For the remaining part :

28

29 Examples:  d

30 Concluding Remarks: 1.Provides precision/error control for all tessellation based application of CCSSs 2.Possible disadvantage: might generate relatively large subdivision depth for vicinity of an extra-ordinary vertex already flat enough 3.Possible improvement: use second order norm for extra-ordinary patches

31 Acknowledgement: Research work presented here is supported by NSF (DMS , DMI ).