1 / 24

Bi-3 C 2 Polar Subdivision

Bi-3 C 2 Polar Subdivision. Ashish Myles University of Florida New York University J ö rg Peters University of Florida. Overview: Bicubic C 2 polar subdivision. What is polar subdivision? (increased artistic freedom) Why is curvature continuity difficult? (and what makes it feasible?).

nika
Download Presentation

Bi-3 C 2 Polar Subdivision

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Bi-3 C2 Polar Subdivision Ashish MylesUniversity of FloridaNew York University Jörg PetersUniversity of Florida

  2. Overview: Bicubic C2 polar subdivision • What is polar subdivision?(increased artistic freedom) • Why is curvature continuity difficult?(and what makes it feasible?)

  3. Uniform Bicubic Splines • Quad-grid mesh input • Degree (3, 3) • Curvature continuous (C2) • Well understood,explicit formulas • Mesh refinement

  4. Generalize mesh connectivity valence ≠ 4 Catmull-Clark (C1, unboundedcurvature) polar Bicubic PolarSubdivision –Karciauskas, Peters '08 (C1, bounded curvature)

  5. Why not Catmull-Clark on Polar? Catmull-Clark Bicubic PolarSubdivision

  6. Subdivision on Polar Connectivity Catmull-Clark C1unbounded curvature Bicubic polarsubdivision C1boundedcurvature C2PS continuouscurvature Bicubic + C2 = Surprise!

  7. Why a surprise? • Subdivision theory ([Peters, Reif '08] Subdivision book) • Our contribution: k = 2, q = 3: bicubic, simple masks, C2 at the poles Degree q = 6 needed for C2 at extraordinary points

  8. Subdivision rules • Small masks that depend solely on the connectivity 1-link 2-link

  9. Curvature comparison C2 Jet subdivision (degree (6,5)) C1 bicubic polar subdivision C2PS (degree (3,3))

  10. Curvature comparison – finger C1 bicubic polar subdivision C2 Jet (6,5) subdivision C2PS (3,3)

  11. Shape – C2PS

  12. Part II • What is polar subdivision?(increased artistic freedom) • Why is curvature continuity difficult?(and what makes it feasible?)

  13. z-coordinate quadratic in x and y coordinates need deg 6! need deg 2 deg 3 deg 1 Why deg 6 for C2 at the pole? • C2 & flexible ⇒ can reconstruct a paraboloid f(x, y) = (x, y, x2 + y2) E.g. polar valence = 6

  14. curve C2PS need deg 2 curve deg 1 Infinite valence • Valence → ∞ ⇒ control points → curves⇒ polar coordinates: f(x, y) = (x, y, x2 + y2)f(r cos(t), r sin(t)) = (r cos(t), r sin(t), r2) valence =∞

  15. Infinite valence ⇒ C2 f(x, y) = (x, y, x2 + y2) = (r cos(t), r sin(t), r2) f(x, y) = (x, y, x2 – y2) = (r cos(t), r sin(t), r cos(2t)) f(x, y) = (x, y, 2 x y) = (r cos(t), r sin(t), r sin(2t)) span{x2 + y2, x2 – y2, 2 x y} = span{x2, y2, x y} = span{x,y} × span{x,y}

  16. ? ? ? ? ? ? ? ? ? Extraordinary neighborhood ... ...

  17. A (5n+1) × (5n+1) matrix q0 q1 5n+1 vertices 5n+1 vertices Recall: Stationary subdivision analysis • q1 = A q0 • Radial-only subdivision ⇒ A is square • q∞ = A∞q0 → eigenanalysis of A

  18. (5(2n)+1)×(5n+1) matrix A Standard theory not applicable to C2PS • q1 = A q0 • A is not square • q∞ = (...AAAA) q0 → Cannot use eigenanalysis • But: valence → ∞ q0 q1 5n+1 vertices 5(2n)+1 vertices

  19. (5(2n)+1)×(5n+1) matrix A Trick – reformulate C2PS • Reformulate C2PS refinement in terms of 6 variables f(x, y) = p0 + p1x + p2y + p3 (x2 + y2) + p4 (x2 – y2) + p5 (2 x y) + o(x2 + y2) • C2PS = Diminishing perturbation of subdivision on infinite valence ⇒C2 q0 q1 5n+1 vertices 5(2n)+1 vertices

  20. Future direction • C2 for polar and regular – how about extraordinary points? • Approx. double the extraordinary facet neighborhood • Can we make the subdivision masks easy too? Catmull-Clark non-stationary refinement

  21. Summary • Bicubic C2 Polar subdivision • simple, curvature continuous • Analysis technique • compare with stationary algorithm(on curves) • Non-stationary valence/connectivity • allows for low degree + high smoothness

  22. Acknowledgments • SIGGRAPHcommittee andreviewers • NationalScienceFoundation(0728797)

  23. Thank you

  24. Thank you

More Related