Boolean Operations on Subdivision Surfaces

1. University of Burgundy Boolean Operations on Subdivision Surfaces Yohan FOUGEROLLE MS 2001/2002 Sebti FOUFOU Marc Neveu

2. Introduction A B A  B A - B B - A A  B

3. Introduction Intersection is needed to deduce other boolean operations Sphere  Cube Sphere  Cube Cube - Sphere Sphere - Cube

4. : Control Points : Mix functions (triangular B-Splines) Subdivision Surfaces • Subdivision Surfaces as NURBS Alternative • Now very used in CAD and animation movies (Geri’s Game, Monster Inc…) • Arbitrary Meshes • Easy patches • Simple use with small datas • Numerous subdivision rules with different properties • Work on Triangular parametric domain

5. 1 3 1 1 1 1 1 1 3 Vertex Mask Edge Mask with LOOP Scheme Vi ,1 Vi ,6 Uniform Approximating scheme Vi +1,1 Vi +1,6 Vi +1,2 Vi ,5 VR Vi +1,5 Vi ,2 Vi +1,3 Vi +1,4 Vi ,3 Vi ,4 New Control Points inserted Each face generates 4 faces

6. Loop Surfaces Example Surface evolution with subdivision level Limit surface

7. « Wrong » Intersections  General problem : No location/existence criterion Subdivision(s) Current Control Mesh Initial mesh Subdivision(s)

8. Intersection Approximation No suitable mathematical criterion Approximation to level N  N subdivisions  Intersection(s) curve(s) Adaptative subdivision to refine the result

9. A A C  A A∩B A C  A Surfaces splitting • Two steps : • Split along the intersection curve • labelling to separate each part of the object (inside/outside the other object) A C

10. Reconstruction • Depending on boolean operation : • Faces are stored in the result object • Merging operation along the intersection curve

11. Example

12. Intersection curve example

13. Splitting and labelling operations Interior faces Exterior faces

14. Results intersection Union Sphere - Torus Torus- Sphere

15. subdivision subdivision one point / edge Adaptative Subdivision Intersection curve

16. Mesh updating Update all on triangular faces With barycenter triangulation

17. Example of adaptative subdivision Approximate Boolean Operations on Free-Form Solids Biermann, Kristjanson, Zorin CAGD Oslo 2000

18. Future works • Minimize the surface perturbations due to adaptative subdivision and triangulation. • Update the intersection algorithm to manage non triangular (planar) faces. • Use a hierarchy data structure ( tree ) to store faces and decrease the intersection algorithm complexity. • Reverse the process to store a smaller mesh.

19. Conclusion • Geometrical approach of intersection  one domain is needed to compute boolean operation. • Works with non convex 3D objects and 2-manifold. • One restriction : an edge must always separate two faces at most.

20. Questions ?