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