A Quadrilateral Rendering Primitive

1. Nira Dyn • Michael Floater • Kai Hormann Dual 2n-Point Schemes A QuadrilateralRendering Primitive

2. Introduction 1 6 1 4 4 • Primal schemes • one new vertex for each old vertex • one new vertex for each old edge • “keep old points, add edge midpoints” • mask with odd length Dual 2n-Point Schemes

3. Introduction 1 3 3 1 • Dual schemes • one new edge for each old vertex • one new edge for each old edge • “add two edge-points, forget old points” • mask with even length Dual 2n-Point Schemes

4. Known Schemes 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 -1 1 0 6 15 9 16 20 15 9 6 0 1 -1 3 0 -25 0 150 256 150 0 -25 0 3 Primal B-Splines Dual linear quadratic cubic quartic quintic 2n-Point 4-point ? 6-point Dual 2n-Point Schemes

5. Primal 2n-Point Schemes -1 0 9 16 9 0 -1 3 0 -25 0 150 256 150 0 -25 0 3 interpolation cubic precision cubic sampling 1 1 1 1 interpolation 9/16 quintic precision quintic sampling -1/16 Dual 2n-Point Schemes

6. Dual 4-Point Scheme cubic sampling -7 105 35 -5 cubic precision cubic sampling -5 35 105 -7 1 1 1 1 105/128 105/128 35/128 35/128 -5/128 -7/128 -7/128 -5/128 Dual 2n-Point Schemes

7. Dual 4-Point Scheme -5 3 34 34 3 -5 -5 8 26 8 -5 -5 13 13 -5 -5 18 -5 ⇒ scheme is O(h4) and symbol contains (1+z)4 a(z) = cubic precision -5 -7 35 105 105 35 -7 -5 = ·(1+z) -5 -2 37 68 37 -2 -5 = ·(1+z)2 = ·(1+z)3 = ·(1+z)4 = ·(1+z)5 ⇒ scheme could be C4 and 4µspan {(x-j)} Dual 2n-Point Schemes

8. Smoothness Analysis -5 3 34 34 3 -5 -5 -5 -5 8 8 8 26 26 26 8 8 8 -5 -5 -5 25 -40 -170 24 103 272 -596 272 103 24 -170 -40 25 • scheme is notC3 a(z) = -5 -7 35 105 105 35 -7 -5 ⇒ C0 |■| = 72/128 < 1 |■| = 84/128 < 1 -5 -2 37 68 37 -2 -5 ⇒ C1 |■| = 42/64 < 1 |■| = 42/64 < 1 ⇒ C2 |■| = 36/32 > 1 2 × |■| = 336/1024 < 1 |■| = 336/1024 < 1 |■| = 256/1024 < 1 |■| = 936/1024 < 1 Dual 2n-Point Schemes

9. Subdivision Matrix -7 105 35 -5 0 0 -5 35 105 -7 0 0 -5 -7 35 105 105 35 -7 -5 0 -7 105 35 -5 0 0 -5 35 105 -7 0 0 0 -7 105 35 -5 0 0 -5 35 105 -7 • right and left eigenvector for 0: 0 = 1 1 = 1/2 2 = 1/4 S = /128 ⇒ 3 = 1/8 4 = 1/16 5 = 9/64 x0 = [1, 1, 1, 1, 1, 1] y0 = [1, -27, 218, 218, -27, 1]/384 Dual 2n-Point Schemes

10. Limit Function -5 -866 3509 43876 3509 -866 -5 49152 49152 49152 49152 49152 49152 49152 1 -27 218 218 -27 1 0 0 384 384 384 384 384 384 • support size 7 • quasi-interpolation Q = I+R = I+I-T [5, 866, -3509, 54428, -3509, 866, 5] / 49152 Dual 2n-Point Schemes

11. Limit Function  ′ ″ Dual 2n-Point Schemes Dual 2n-Point Schemes

12. Dual 4-Point Scheme • Summary • reproduces cubic polynomials • approximation order O(h4) • C2 continuous • support size 7 • contains quartic polynomials Dual 2n-Point Schemes

13. 2n-Point-Schemes 1 2 1 -5 -7 35 105 105 35 -7 -5 -1 0 9 16 9 0 -1 1 3 3 1 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 3 0 -25 0 150 256 150 0 -25 0 3 Primal 2n-Point Dual 2-Point linear 4-Point cubic 6-Point quintic ∶ ∶ Dual 2n-Point Schemes

14. Dual 4-Point Scheme Dual 2n-Point Schemes

15. Dual 6-Point Scheme Dual 2n-Point Schemes

16. Dual 8-Point Scheme Dual 2n-Point Schemes

17. Examples Dual 2n-Point Schemes

18. Examples Dual 2n-Point Schemes

19. Thank You for Your Attention Dual 2n-Point Schemes A QuadrilateralRendering Primitive