Boolean Operations on Subdivision Surfaces Yohan FOUGEROLLE MS

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

Introduction A A B B A-B B-A A B

Introduction Intersection is needed to deduce other boolean operations Sphere Cube - Sphere - Cube

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 BSplines)

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

Loop Surfaces Example Surface evolution with subdivision level Limit surface

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

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

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

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

Example

Intersection curve example

Splitting and labelling operations Interior faces Exterior faces

Results intersection Union Sphere - Torus- Sphere

Adaptative Subdivision Intersection curve subdivision one point / edge

Mesh updating Update all on triangular faces With barycenter triangulation

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

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.

Conclusion ØGeometrical approach of intersection one domain is needed to compute boolean operation. ØWorks with non convex 3 D objects and 2 -manifold. ØOne restriction : an edge must always separate two faces at most.