SLIDE 1
Boolean Operations
- n Subdivision
Surfaces
Yohan FOUGEROLLE MS 2001/2002 Sebti FOUFOU Marc Neveu
University of Burgundy
Introduction
A B A ∩ B A ∪ B A - B B - A
Boolean Operations on Subdivision Surfaces Yohan FOUGEROLLE MS - - PowerPoint PPT Presentation
Boolean Operations on Subdivision Surfaces Yohan FOUGEROLLE MS - - PowerPoint PPT Presentation
Boolean Operations on Subdivision Surfaces Yohan FOUGEROLLE MS 2001/2002 Sebti FOUFOU University of Marc Neveu Burgundy Introduction A B A B A - B B - A A B Introduction Intersection is needed to deduce other Sphere
SLIDE 2
SLIDE 3
Introduction
Intersection is needed to deduce other boolean operations
Sphere ∩ Cube Sphere ∪ Cube Sphere - Cube Cube - Sphere
SLIDE 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
β , I
C
I
N β
: Control Points : Mix functions (triangular B- Splines)
SLIDE 5
LOOP Scheme
n n V V V n V
r n r r r
+ + + + =
+
) ( ... * ) (
2 1 1
α α
8 * 3 * 3
1 1 1 r i r i r i r r i
V V V V V
+ − +
+ + + =
Vertex Mask 1
) (n α
1 1 1 1 Edge Mask 3 3 1 1 ) ( )) ( 1 ( ) ( n a n a n n − = α 64 2 cos * 2 3 8 5 ) (
2
+ − = n n a π with New Control Points inserted Each face generates 4 faces
Uniform Approximating scheme Vi ,6 Vi ,5 Vi ,4 Vi ,1 Vi ,3 Vi ,2
Vi +1,6 Vi +1,5 Vi +1,4 Vi +1,3 Vi +1,1 Vi +1,2
VR
SLIDE 6
Loop Surfaces Example
Surface evolution with subdivision level Limit surface
SLIDE 7
« Wrong » Intersections
General problem : No location/existence criterion
Subdivision(s) Subdivision(s) Initial mesh Current Control Mesh
SLIDE 8
Intersection Approximation
No suitable mathematical criterion Approximation to level N N subdivisions Intersection(s) curve(s) Adaptative subdivision to refine the result
SLIDE 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
A A C ⊂ A A C ⊂ A C A
SLIDE 10
Reconstruction
Depending on boolean operation : Faces are stored in the result object Merging operation along the intersection curve
SLIDE 11
Example
SLIDE 12
Intersection curve example
SLIDE 13
Splitting and labelling
- perations
Interior faces Exterior faces
SLIDE 14
Results
intersection Union Sphere - Torus Torus- Sphere
SLIDE 15
Adaptative Subdivision
- ne point / edge
subdivision subdivision
Intersection curve
SLIDE 16
Mesh updating
Update all on triangular faces With barycenter triangulation
SLIDE 17
Example of adaptative subdivision
Approximate Boolean Operations on Free-Form Solids Biermann, Kristjanson, Zorin CAGD Oslo 2000
SLIDE 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.
SLIDE 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.
SLIDE 20