boolean operations on subdivision surfaces
DESCRIPTION
University of Burgundy. Boolean Operations on Subdivision Surfaces. Yohan FOUGEROLLE MS 2001/2002 Sebti FOUFOU Marc Neveu. Introduction. A. B. A B. A - B. B - A. A B. Introduction. Intersection is needed to deduce other boolean operations. Sphere Cube. - PowerPoint PPT PresentationTRANSCRIPT
Boolean Operations on
Subdivision Surfaces
Yohan FOUGEROLLEMS 2001/2002Sebti FOUFOU
Marc NeveuUniversity of Burgundy
Introduction
A B
A B A BA - B B - A
Introduction
Intersection is needed to deduce other boolean operations
Sphere Cube
Sphere Cube
Sphere - CubeCube - Sphere
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
,ICIN
: Control Points
: Mix functions (triangular B-Splines)
LOOP Scheme
nn
VVVnV
rn
rrr
)(
...*)( 211
8
*3*3 111ri
ri
ri
rri
VVVVV
Vertex Mask
1
)(n1
1 1
1
Edge Mask
3
3
11
)(
))(1()(
na
nann
64
2cos*23
8
5)(
2
n
na
with
New Control Points inserted
Each face generates 4 faces
Uniform Approximating schemeVi ,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
Loop Surfaces Example
Surface evolution with subdivision level
Limit surface
« Wrong » Intersections
General problem : No location/existence criterion
Subdivision(s)
Subdivision(s)Initial mesh
Current Control Mesh
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
labelling to separate each part of the object (inside/outside the other object)
A∩B
A A C A
A C A CA
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 Torus- Sphere
Adaptative Subdivision
one point / edge
subdivision subdivision
Intersection curve
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
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.