Transcript Subdivision Overview Control Mesh Topological Split • Subdivision is a two part process – Topological split – Local averaging / smoothing Averaging.

Subdivision Overview
Control Mesh
Topological Split
• Subdivision is a two part process
– Topological split
– Local averaging / smoothing
Averaging
Subdivision Overview
Control Mesh
Generation 1
Generation 2
Generation 3
• Repeated uniform subdivisions of the control mesh
converge to the limit surface
• Stationary schemes (averaging mask does not change)
– Limit position and normal from eigen-analysis
Bspline Surfaces
• A single Bspline surface patch is controlled
by a regular 4x4 grid of control points
Bspline Surfaces
• 2 adjacent patches share 12 control points
and meet with C2 continuity
Bspline Surfaces
• Requires regular rectangular (toroidal) control mesh
to guarantee continuity (all valence-4 vertices!)
• Subdivision can be performed by knot insertion
(i.e. blossoming)
Catmull-Clark Subdivision
Surfaces
• Smooth surfaces for control meshes
of arbitrary topology
– Closed control mesh
 closed limit surface
• Quad mesh generalization of Bsplines
– C1 at non-valence-4 vertices,
– C2 every where else (Bsplines).
• Sharp corners can be tagged:
– allows for smooth and sharp features;
– allows for non-closed meshes.
Catmull-Clark Subdivision
Gen 0
Gen 1
Gen 2
• Extraordinary vertices are generated by
non-valence-4 vertices & faces in the input mesh.
• No further extraordinary vertices are created
after the first generation of subdivision.
Catmull-Clark Averaging
C20
NEW EDGE POINTS:
V20
C30
V3 0
F2 0
F3
V4
C10
0
E20
E30
E40 V00
F1 0
E10
0
F4
0
V10
En0
Fn 0
C40
Vn
0
Cn
0
g
g
V

V
i
E ig  0
2
NEW FACE POINTS:
g
g
g
g
V

V

C

V
i
i
i 1
Fig  0
4
SMOOTHED VERTICES:
1 n g 1 n g
(n - 3)V  2  E i   Fi
n i 1
n i 1
g 1
V0 
n
g
0
Loop Subdivision Surfaces
Gen 0
NEW EDGE POINTS:
g
g
V

V
i
E ig  0
2
Gen 1
Gen 2
2

3  2 cos(2 / n)
a
32
n
1
V0g 1  aV0g  (1  a)  E ig
n i 1