1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve...
-
date post
21-Dec-2015 -
Category
Documents
-
view
214 -
download
1
Transcript of 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve...
![Page 1: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/1.jpg)
1
Dr. Scott Schaefer
Subdivision Curves
![Page 2: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/2.jpg)
2/96
What is subdivision?
Set of rules S that take a curve as input and produce a more highly refined curve as output
Recursively applying S yields a sequence of curves which should converge to some limit shape
S S
![Page 3: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/3.jpg)
3/96
Subdivision Rules
Typically chosen to be linear combinations of neighboring vertices
Rules usually depend only on local topology of shape
Reposition Old Vertices Insert New Vertices
43
81
81
21
21
![Page 4: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/4.jpg)
4/96
Applying Subdivision Rules
![Page 5: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/5.jpg)
5/96
Applying Subdivision Rules
21
21
![Page 6: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/6.jpg)
6/96
Applying Subdivision Rules
81
43
81
![Page 7: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/7.jpg)
7/96
Applying Subdivision Rules
21
21
![Page 8: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/8.jpg)
8/96
Applying Subdivision Rules
81
81 4
3
![Page 9: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/9.jpg)
9/96
Applying Subdivision Rules
21
21
![Page 10: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/10.jpg)
10/96
Applying Subdivision Rules
81
81
43
![Page 11: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/11.jpg)
11/96
Applying Subdivision Rules
21
21
![Page 12: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/12.jpg)
12/96
Applying Subdivision Rules
81
81
43
![Page 13: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/13.jpg)
13/96
Applying Subdivision Rules
![Page 14: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/14.jpg)
14/96
Applying Subdivision Rules
![Page 15: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/15.jpg)
15/96
Applying Subdivision Rules
![Page 16: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/16.jpg)
16/96
Subdivision Rules Via Blossoming
Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,…
![Page 17: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/17.jpg)
17/96
)4,2,0(b
t6 t8 t100t 4t2t
t6 2t t8 4t
),,( tttb
t6 4t
),4,2( tb ),6,4( tb ),8,6( tb
),,4( ttb ),,6( ttb
)6,4,2(b )8,6,4(b )10,8,6(b
Subdivision Rules Via Blossoming
Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,…
)4,2,0(b)6,4,2(b)8,6,4(b)10,8,6(b
Given)5,4,3(b)6,5,4(b
Find
![Page 18: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/18.jpg)
18/96
21
21
)6,5,4(b
)6,4,2(b )8,6,4(b
Subdivision Rules Via Blossoming
Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,…
)4,2,0(b)6,4,2(b)8,6,4(b)10,8,6(b
Given)5,4,3(b)6,5,4(b
Find
![Page 19: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/19.jpg)
19/96
)4,2,0(b
21
65
21
61
41
43
)4,3,2(b )6,4,3(b
)5,4,3(b
)6,4,2(b )8,6,4(b
Subdivision Rules Via Blossoming
Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,…
)4,2,0(b)6,4,2(b)8,6,4(b)10,8,6(b
Given)5,4,3(b)6,5,4(b
Find
![Page 20: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/20.jpg)
20/96
Subdivision Rules Via Blossoming
Assume knot-spacing uniform 0,2,4,6,8,… Find control points for refined knots 0,1,2,3,…
)4,2,0(b)6,4,2(b)8,6,4(b)10,8,6(b
Given)8,6,4()6,4,2()4,2,0()5,4,3( 8
143
81 bbbb
)8,6,4()6,4,2()6,5,4( 21
21 bbb
Find
Works for arbitrary degree B-splines!!!
![Page 21: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/21.jpg)
21/96
Lane Reisenfeld Subdivision
Linearly subdivide the curve by inserting the midpoint on each edge
Perform averaging by replacing each edge by its midpoint d times
![Page 22: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/22.jpg)
22/96
Lane Reisenfeld Subdivision
0p 1p 2p210 pp
221 pp
232 pp
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
![Page 23: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/23.jpg)
23/96
Lane Reisenfeld Subdivision
![Page 24: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/24.jpg)
24/96
Lane Reisenfeld Subdivision
![Page 25: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/25.jpg)
25/96
Lane Reisenfeld Subdivision
![Page 26: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/26.jpg)
26/96
Lane Reisenfeld Subdivision
![Page 27: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/27.jpg)
27/96
Lane Reisenfeld Subdivision
![Page 28: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/28.jpg)
28/96
Lane Reisenfeld Subdivision
![Page 29: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/29.jpg)
29/96
Lane Reisenfeld Subdivision
![Page 30: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/30.jpg)
30/96
Lane Reisenfeld Subdivision
![Page 31: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/31.jpg)
31/96
Interpolatory Subdivision
Interpolating control vertices may be desirable
Catmull-Rom splines are not refinable!!!
Reposition Old Vertices Insert New Vertices
1
00
? ?
??
![Page 32: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/32.jpg)
32/96
Four-Point Subdivision
0p
1p
2p
3p 4p
0 1 2 3t
4
![Page 33: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/33.jpg)
33/96
Four-Point Subdivision
0p
1p
2p
3p 4p
0 1 2 3t
4
![Page 34: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/34.jpg)
34/96
Four-Point Subdivision
0p
1p
2p
3p 4p
0 1 2 3t
4
![Page 35: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/35.jpg)
35/96
Four-Point Subdivision
3
2
1
0
3
2
1
0
33
233
32
222
31
211
30
200
1
1
1
1
p
p
p
p
c
c
c
c
ttt
ttt
ttt
ttt
0p
1p
2p
3p
0 1 2 3 t
![Page 36: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/36.jpg)
36/96
Four-Point Subdivision
3
2
1
0
3
2
1
0
32
32
32
27931
8421
1
0001
p
p
p
p
c
c
c
c
0p
1p
2p
3p
0 1 2 3 t
![Page 37: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/37.jpg)
37/96
Four-Point Subdivision
3
2
1
0
1
32
32
32
3
2
1
0
27931
8421
1
0001
p
p
p
p
c
c
c
c
0p
1p
2p
3p
0 1 2 3 t
![Page 38: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/38.jpg)
38/96
Four-Point Subdivision
3
2
1
0
1
32
32
323
8272
49
23
3
2
1
0
38272
49
23
27931
8421
1
0001
11
p
p
p
p
c
c
c
c
0p
1p
2p
3p
0 1 2 3 t
![Page 39: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/39.jpg)
39/96
Four-Point Subdivision
3
2
1
0
161
169
169
161
p
p
p
p
0p
1p
2p
3p
0 1 2 3 t
![Page 40: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/40.jpg)
40/96
Four-Point Subdivision
![Page 41: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/41.jpg)
41/96
Four-Point Subdivision
![Page 42: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/42.jpg)
42/96
Four-Point Subdivision
169
169
161
161
![Page 43: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/43.jpg)
43/96
Four-Point Subdivision
161
169
161
169
![Page 44: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/44.jpg)
44/96
Four-Point Subdivision
161
161
169
169
![Page 45: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/45.jpg)
45/96
Four-Point Subdivision
169
161
169
161
![Page 46: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/46.jpg)
46/96
Four-Point Subdivision
![Page 47: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/47.jpg)
47/96
Four-Point Subdivision
![Page 48: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/48.jpg)
48/96
Four-Point Subdivision
![Page 49: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/49.jpg)
49/96
Subdivision as Basis Function Refinement
i
ipixNxp )()(
![Page 50: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/50.jpg)
50/96
Subdivision as Basis Function Refinement
i
ipixNxp )()(
)(xN
Cubic B-spline Basis Function
![Page 51: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/51.jpg)
51/96
Subdivision as Basis Function Refinement
i
ipixNxp )()(
)(xN
Four-Point Basis Function
![Page 52: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/52.jpg)
52/96
Subdivision as Basis Function Refinement
i
ipixNxp )()(
1
0
1
...)1()()1(...)(
p
p
p
xNxNxNxp
![Page 53: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/53.jpg)
53/96
Subdivision as Basis Function Refinement
i
ipixNxp )()(
1
0
1
81
43
81
21
21
81
43
81
21
21
81
43
81
00
000
...00...
000
00
...)12()2()12(...)(
p
p
p
xNxNxNxp
![Page 54: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/54.jpg)
54/96
Subdivision as Basis Function Refinement
1
0
1
81
43
81
21
21
81
43
81
21
21
81
43
81
00
000
...00...
000
00
...)12()2()12(...)(
p
p
p
xNxNxNxp
1
0
1
...)1()()1(...)(
p
p
p
xNxNxNxp
![Page 55: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/55.jpg)
55/96
Subdivision as Basis Function Refinement
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
![Page 56: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/56.jpg)
56/96
Subdivision as Basis Function Refinement
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
![Page 57: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/57.jpg)
57/96
Subdivision as Basis Function Refinement
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
43
21
81
81
21
![Page 58: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/58.jpg)
58/96
Subdivision as Basis Function Refinement
)32()12()2()12()32()( 161
169
169
161 xNxNxNxNxNxN
![Page 59: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/59.jpg)
59/96
Subdivision as Basis Function Refinement
)32()12()2()12()32()( 161
169
169
161 xNxNxNxNxNxN
1
169
169
161
161
![Page 60: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/60.jpg)
60/96
Limit Points of Curve Subdivision
0p 1p 2p2p 1p
i
ipixNxp )()(
![Page 61: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/61.jpg)
61/96
Limit Points of Curve Subdivision
0p 1p 2p2p 1p
i
ipiNp )()0(
![Page 62: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/62.jpg)
62/96
Limit Points of Curve Subdivision
0p 1p 2p2p 1p
),...2(),1(),0(),1(),2(..., NNNNN
![Page 63: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/63.jpg)
63/96
Limit Points of Curve Subdivision
0p 1p 2p2p 1p
1)()()( ii
i ixNpixNxp
![Page 64: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/64.jpg)
64/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
43
21
81
81
21
1)( i
ixN
![Page 65: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/65.jpg)
65/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
1)( i
ixN
)2()1()0()1()2()0( 81
21
43
21
81 NNNNNN
![Page 66: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/66.jpg)
66/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
1)( i
ixN
)2()1()0()1()2()0( 81
21
43
21
81 NNNNNN
Symmetry!!!
![Page 67: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/67.jpg)
67/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
1)( i
ixN
)2()1()0()0( 41
43 NNNN
![Page 68: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/68.jpg)
68/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
1)( i
ixN
)2()1()0()0( 41
43 NNNN
Zero!!!
0 1 2 33 2 1
![Page 69: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/69.jpg)
69/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
1)( i
ixN
)1()0()0( 43 NNN
![Page 70: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/70.jpg)
70/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
1)( i
ixN
)1()0()0( 43 NNN
)4()3()2()1()0()1( 81
21
43
21
81 NNNNNN
![Page 71: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/71.jpg)
71/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
1)( i
ixN
)1()0()0( 43 NNN
)1()0()1( 21
81 NNN
![Page 72: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/72.jpg)
72/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
1)( i
ixN
)1()0()0( 43 NNN
)1()0()1( 21
81 NNN
1)1(2)0( NN
![Page 73: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/73.jpg)
73/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
1)( i
ixN
)1()0()0( 43 NNN
)1()0()1( 21
81 NNN
1)1(2)0( NN
0 1 2 33 2 1
32
61
61
![Page 74: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/74.jpg)
74/96
Limit Points of Curve Subdivision
0 1 2 33 2 1
i
ipiNp )()( 31
31
![Page 75: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/75.jpg)
75/96
Limit Points of Curve Subdivision
0 1 2 33 2 1
),...(),(),(),(),(..., 35
32
31
34
37 NNNNN
![Page 76: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/76.jpg)
76/96
Evaluate scaling relationship at Solve linear system of equations with
constraint
Limit Points of Curve Subdivision
niN
32
5431
2710
61
814
1621
0 1 2 33 2 1
1Zd
ni dN
![Page 77: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/77.jpg)
77/96
Limit Points of Curve Subdivision
)32()12()2()12()32()( 161
169
169
161 xNxNxNxNxNxN
1
55894240
55892000
0
5589410
5589256
0558916
55891
![Page 78: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/78.jpg)
78/96
Limit Points of Curve Subdivision
Assume N(x) has finite support Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
![Page 79: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/79.jpg)
79/96
Limit Points of Curve Subdivision
Assume N(x) has finite support Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
0)22()( yNyN
![Page 80: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/80.jpg)
80/96
Limit Points of Curve Subdivision
Assume N(x) has finite support Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
0)22()( yNyN
22 yy
![Page 81: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/81.jpg)
81/96
Limit Points of Curve Subdivision
Assume N(x) has finite support Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
0)22()( yNyN
2y
![Page 82: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/82.jpg)
82/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
)2()1()0()1()2()0( 81
21
43
21
81 NNNNNN
)4()3()2()1()0()1( 81
21
43
21
81 NNNNNN
)0()1()2()3()4()1( 81
21
43
21
81 NNNNNN
![Page 83: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/83.jpg)
83/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
)1()0()1()0( 21
43
21 NNNN
)1()0()1( 21
81 NNN
)0()1()1( 81
21 NNN
![Page 84: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/84.jpg)
84/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
)1()0()1()0( 21
43
21 NNNN
)1()0()1( 21
81 NNN
)0()1()1( 81
21 NNN
21
21
81
43
81
21
21
0
0
)1(),0(),1()1(),0(),1( NNNNNN
![Page 85: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/85.jpg)
85/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
)1()0()1()0( 21
43
21 NNNN
)1()0()1( 21
81 NNN
)0()1()1( 81
21 NNN
21
21
81
43
81
21
21
0
0
)1(),0(),1()1(),0(),1( NNNNNN
Limit mask is left-eigenvector corresponding to 1
![Page 86: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/86.jpg)
86/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
)1()0()1()0( 21
43
21 NNNN
)1()0()1( 21
81 NNN
)0()1()1( 81
21 NNN
1,4,1)1(),0(),1( NNN
![Page 87: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/87.jpg)
87/96
Limit Points of Curve Subdivision
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
)1()0()1()0( 21
43
21 NNNN
)1()0()1( 21
81 NNN
)0()1()1( 81
21 NNN
61
32
61 ,,)1(),0(),1( NNN
1)( i
ixN
![Page 88: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/88.jpg)
88/96
Derivatives of Subdivision Curves
)22()12()2()12()22()( 81
21
43
21
81 xNxNxNxNxNxN
![Page 89: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/89.jpg)
89/96
Derivatives of Subdivision Curves
)22(')12(')2(')12(')22('2)(' 81
21
43
21
81 xNxNxNxNxNxN
![Page 90: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/90.jpg)
90/96
Derivatives of Subdivision Curves
)22(')12(')2(')12(')22('2)(' 81
21
43
21
81 xNxNxNxNxNxN
21
21
81
43
81
21
21
21
0
0
)1('),0('),1(')1('),0('),1(' NNNNNN
![Page 91: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/91.jpg)
91/96
Derivatives of Subdivision Curves
)22(')12(')2(')12(')22('2)(' 81
21
43
21
81 xNxNxNxNxNxN
21
21
81
43
81
21
21
21
0
0
)1('),0('),1(')1('),0('),1(' NNNNNN
left-eigenvector corresponding to 1/2
![Page 92: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/92.jpg)
92/96
Derivatives of Subdivision Curves
)22(')12(')2(')12(')22('2)(' 81
21
43
21
81 xNxNxNxNxNxN
1,0,1)1('),0('),1(' NNN
![Page 93: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/93.jpg)
93/96
Four-Point Limits and Derivatives
)32()12()2()12()32()( 161
169
169
161 xNxNxNxNxNxN
![Page 94: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/94.jpg)
94/96
Four-Point Limits and Derivatives
0,0,1,0,0)2(),1(),0(),1(),2( NNNNN
)32()12()2()12()32()( 161
169
169
161 xNxNxNxNxNxN
![Page 95: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/95.jpg)
95/96
Four-Point Limits and Derivatives
0,0,1,0,0)2(),1(),0(),1(),2( NNNNN
)32()12()2()12()32()( 161
169
169
161 xNxNxNxNxNxN
01000
0
00100
0
00010
)2('),1('),0('),1('),2(')2('),1('),0('),1('),2('
161
169
169
161
161
169
169
161
21 NNNNNNNNNN
![Page 96: 1 Dr. Scott Schaefer Subdivision Curves. 2/96 What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d6d5503460f94a4ceda/html5/thumbnails/96.jpg)
96/96
Four-Point Limits and Derivatives
0,0,1,0,0)2(),1(),0(),1(),2( NNNNN
)32()12()2()12()32()( 161
169
169
161 xNxNxNxNxNxN
1,8,0,8,1)2('),1('),0('),1('),2(' NNNNN