Overview June 9- B-Spline Curves June 16- NURBS Curves June 30- B-Spline Surfaces.
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Transcript of Overview June 9- B-Spline Curves June 16- NURBS Curves June 30- B-Spline Surfaces.
)()(),( vBuBvuB jiij Tensor Product B-Spline
iu 1iu 2iu 3iu 4iu
iu 1iu2iu 3iu
4iu
jv1jv
2jv3jv
4jv
z
)(uBi
),( vuBij2C
),( vuBij-Piecewise (16 parametric regions);
-Bicubic;
-C2;
-Local Support;
-Local supports cover the plane of the parameters in a regular fashion;
-Sum up to one;
Tensor Product B-Spline
Uniform cubic B-spline Curves
Uniform bicubic B-spline Surfaces
)()( uBu ii
i VQ ),(),( vuBvu iji j
ij VQ
iV ijV
)(uBi )()(),( vBuBvuB jiij
0u
Expression
Vertices
Basis Functions
Parameter Space
1u 2u Mu
0u 1u 2u 3u 4u 1Mu Mu
Mv
1Mv
0v
1v
3v
4v
)()(),(),(0
3
0
3,,, 11
vBuBvuvu sjrir s
sjrivvuuijjjii
VQQ
),( vuijQ
),()(0
3
0
3, vbub sr
r ssjri
V
10,10 vu
Local expression
u
Surfaces and Curves
),()()()( 03122130 vbvbvbvb WWWW
)()()(0
3
0
3,,, vbubv sr
r ssjriuji VQ
);()()()( 0,1,12,23,33 ubububub jijijiji VVVVW
);()()()( 01,11,121,231,32 ubububub jijijiji VVVVW
);()()()( 02,12,122,232,31 ubububub jijijiji VVVVW
).()()()( 03,13,123,233,30 ubububub jijijiji VVVVW
Continuity: C2
)()()()()( 04132231,1, vbvbvbvbvuji WWWWQ
).()()()( 01,11,121,23!,34 ubububub jijijiji VVVVW
ujv
1jv
iu 1iu
2jv
3,3,13,23,3
2,2,12,22,3
1,1,11,21,3
,,1,2,3
jijijiji
jijijiji
jijijiji
jijijiji
VVVV
VVVV
VVVV
VVVV
Surface Patch
3,13,3,13,23,3
2,12,2,12,22,3
1,11,1,11,21,3
,1,,1,2,3
1,11,1,11,21,3
jijijijiji
jijijijiji
jijijijiji
jijijijiji
jijijijiji
VVVVV
VVVVV
VVVVV
VVVVV
VVVVV
Four Patches
- C2
- Counting the surface patches…
- Convex Hull
- Rotation
- Scaling
- Translation
It requires 16 Control Vertices to define a patch.
11 nm Control Vertices
22 nm Patches
Properties: Uniform bicubic B-spline Surfaces
),( vuijQ )()(0
3
0
3, vbub sr
r ssjri
V
Boundary Conditions
3,3,13,23,3
2,2,12,22,3
1,1,11,21,3
,,1,2,3
jijijiji
jijijiji
jijijiji
jijijiji
VVVV
VVVV
VVVV
VVVV
3,3,3,13,23,33,3
3,3,3,13,23,33,3
2,2,2,12,22,32,3
1,1,1,11,21,31,3
,,,1,2,3,3
,,,1,2,3,3
jijijijijiji
jijijijijiji
jijijijijiji
jijijijijiji
jijijijijiji
jijijijijiji
VVVVVV
VVVVVV
VVVVVV
VVVVVV
VVVVVV
VVVVVV
3,3,3,3,13,23,33,33,3
3,3,3,3,13,23,33,33,3
3,3,3,3,13,23,33,33,3
2,2,2,2,12,22,32,32,3
1,1,1,1,11,21,31,31,3
,,,,1,2,3,3,3
,,,,1,2,3,3,3
,,,,1,2,3,3,3
jijijijijijijiji
jijijijijijijiji
jijijijijijijiji
jijijijijijijiji
jijijijijijijiji
jijijijijijijiji
jijijijijijijiji
jijijijijijijiji
VVVVVVVV
VVVVVVVV
VVVVVVVV
VVVVVVVV
VVVVVVVV
VVVVVVVV
VVVVVVVV
VVVVVVVV
Generalization- Tensor Product Surfaces
)()(0 uBuB m
)(
)(0
vB
vB
m
mnm
n
VV
VV
0
000
),( vuQ
-Choice of basic functions;
-Given the vertices, we may compute the approximation surface;
-Given a set of points in the surface, we can compute the vertices of the interpolating surface.
Tensor Product Interpolants
)ˆ()ˆ(
)ˆ()ˆ(
)ˆ()ˆ(
)ˆ()ˆ(
0
000
0
000
0
000
mmm
m
mnm
n
mmm
m
vBvB
vBvB
uBuB
uBuB
VV
VV
)ˆ,ˆ()ˆ,ˆ(
)ˆ,ˆ()ˆ,ˆ(
0
000
mmm
m
vuvu
vuvu
Given
Wanted
Triangular Patch Surfaces
Barycentric Coordinates (r,s,t)
),,(),( ,,,,
,, tsrBvu nkji
nkji
kjikji
bQ
Control Vertices
kji ,,b
0,01,00,1),,( PPPP tsrtsr
Bernstein Polynomials
kjinkji tsr
kji
ntsrB
!!!
!),,(,,
Local Expression of a triangular Bezier Patch
P
0,1P0,0P
1,0P