Download - A Quadrilateral Rendering Primitive
A QuadrilateralRendering Primitive
Nira Dyn • Michael Floater • Kai Hormann
Dual 2n-Point Schemes
Dual 2n-Point Schemes
Primal schemesone new vertex for each old vertexone new vertex for each old edge
“keep old points, add edge midpoints”mask with odd length
Introduction
1 6 1
4 4
Dual 2n-Point Schemes
Dual schemesone new edge for each old vertexone new edge for each old edge
“add two edge-points, forget old points”mask with even length
Introduction
1 3
3 1
Dual 2n-Point Schemes
Known Schemes
1 33 1
1 6 14 4
1 105 10 5 1
1 15 156 20 16
1 12
B-Splines
linear
cubic
quintic
quadratic
quartic
2n-Point
-1 9 90 16 -10
-25 150 1500 256 -250 303 0
4-point
6-point?
Primal Dual
Dual 2n-Point Schemes
quintic precisioninterpolation
Primal 2n-Point Schemes
-1 9 9 -10 16 0
-25 150 150 -25 33 0 256 0 00
interpolation
cubic sampling
1
-1/16
9/161 1 1
quintic sampling
cubic precision
Dual 2n-Point Schemes
Dual 4-Point Scheme
-5 35 105 -7
-7 105 35 cubic sampling
cubic sampling
1
-7/128
1 1 1
cubic precision
-5
-5/128
105/128 35/12835/128 105/128
-7/128-5/128
Dual 2n-Point Schemes
Dual 4-Point Scheme
-5 35 105 -7-7 105 35 cubic precision-5
⇒ scheme is O(h4) and symbol contains (1+z)4
-5 37 37 -5-2 68 -2
-5 34 33 34 -5
-5 26 -58 8
-5 1313 -5
-5 -518
a(z) =
= ·(1+z)
= ·(1+z)2
= ·(1+z)3
= ·(1+z)4
= ·(1+z)5
⇒ scheme could be C4 and 4 µ span {(x-j)}
Dual 2n-Point Schemes
⇒ C2
|■| = 42/64 < 1⇒ C1
Smoothness Analysis
-5 35 105 -7-7 105 35 -5
-5 37 37 -5-2 68 -2
-5 34 33 34 -5
-5 26 -58 8
a(z) =
|■| = 84/128 < 1|■| = 72/128 < 1
⇒ C0
|■| = 42/64 < 1
|■| = 36/32 > 1
25 -170 103-40 24 272 272 24-596 103 -170 25-40
-5 26 -58 8 -5 26 -58 8× 2
|■| = 336/1024 < 1|■| = 256/1024 < 1|■| = 936/1024 < 1|■| = 336/1024 < 1
scheme is not C3
Dual 2n-Point Schemes
right and left eigenvector for 0:
Subdivision Matrix
-5 35 105 -7
-7 105 35 -5
-5 35 105 -7
-7 105 35 -5
-5 35 105 -7
-7 105 35 -5
0 0
0 0
0
0
0
0
0
0
0
0S =
-5 35 105 -7-7 105 35 -5
/128 ⇒
0 = 11 = 1/22 = 1/43 = 1/84 = 1/165 = 9/64
x0 = [1, 1, 1, 1, 1, 1]
y0 = [1, -27, 218, 218, -27, 1]/384
Dual 2n-Point Schemes
Limit Function
support size 7quasi-interpolation Q = I+R = I+I-T
[5, 866, -3509, 54428, -3509, 866, 5] / 49152
-5 -5-866-866 3509 35094387649152491524915249152 49152 49152 49152
0 0218384
218384
-27384
-27384
1384
1384
Dual 2n-Point Schemes
Limit Function
Dual 2n-Point Schemes
′
″
Dual 2n-Point Schemes
Dual 4-Point Scheme
Summary reproduces cubic polynomialsapproximation order O(h4)C2 continuoussupport size 7contains quartic polynomials
Dual 2n-Point Schemes
2n-Point-Schemes
2n-Point
2-Pointlinear
4-Pointcubic
6-Pointquintic
-1 9 90 16 -10
-25 150 1500 256 -250 303 0
Primal Dual
1 12
-5 35 105 -7-7 105 35 -5
1 33 1
⋯ ⋯ ⋯ ⋯⋯ ⋯ ⋯ ⋯⋯ ⋯ ⋯ ⋯
∶ ∶
Dual 2n-Point Schemes
Dual 4-Point Scheme
Dual 2n-Point Schemes
Dual 6-Point Scheme
Dual 2n-Point Schemes
Dual 8-Point Scheme
Dual 2n-Point Schemes
Examples
Dual 2n-Point Schemes
Examples
A QuadrilateralRendering Primitive
Thank You for Your Attention
Dual 2n-Point Schemes
