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

2. Introduction AB A  BA  B A - BB - A

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

4. 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 : Control Points : Mix functions (triangular B- Splines)

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

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

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

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

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

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 intersectionUnionSphere - TorusTorus- Sphere

15. Adaptative Subdivision one point / edge 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.