1. Ribbed Surfacesfor Art, Architecture, Visualization James Hamlin and Carlo H. Séquin University of California, Berkeley Computer-Aided Design and Applications Reno, June 11, 2009

2. Overview • Charles O. Perry's Solstice • Reverse engineering Solstice • Ribbed Surface Paradigm • Solstice Program • Visualization of complex surfaces • Ribbed surfaces in architecture

3. Motivation • Inspired byCharles O. Perry'sribbed sculptures. • Emulate and generalize by abstraction.

5. Parameterization of Solstice • (3, 2) torus knot • Curved “ribs” in nearly • triangular configuration

6. Parameterization of Solstice • (3, 2) torus knot • Curved “ribs” in nearly • triangular configuration

7. Parameterization of Solstice Guide rail: (3, 2) torus knot Staggering of ribs:rib offset alongguide rail Rib shapes: concave “hyperbolic” triangles

8. Ribbed Surfaces • Guide rail(s) • very application specific. • Ribs swept along rail(s) • shapes determined procedurally, • e.g., in terms of guide rail derivative information (Frenet frame). • Reduces the number of input parameters • (e.g., compared to sweep surfaces).

9. Sweep Surfaces A One or two path or rail curves One or two more profile curves Maya: Extrusions (A), Lofts (B), Bi-Rails (C). (A) (B) (C)

10. Sweeping Ribs Single rail [0, 0.5) → [0.5, 1.0) Two rails [0, 0.5] → [0.0, 1.0]

11. Guide Rails: Solstice Guide rails are application-specific For Solstice: ( p, q ) torus knots ( 4, 3 ) ( 2, 3 ) ( 3, 2 )

12. Sweeping Ribs: Solstice Rib Offsets: 0° 83° 303° 360°

13. Rib Parameterization Circular Arcs Embedding plane Turning angle • Cubic Hermite • Tangent directions and magnitudes at both ends Rails Rails θ

14. Cubic Hermite Ribs • End tangents specified in terms of Frenet frames of guide rails. n b t V

15. Symmetric, Planar Cubic Hermite Ribs • Constrain ribs to be symmetric, planar. • Select a plane through chord with an angle against rail tangent. • Rib tangent angles are offset from chord;or a curve offset dfrom chord is set.

16. 3D Cubic Hermite Ribs • A combination of the previous two approaches. • Uses: rail tangent,chord direction, and their cross product.

17. Rib Shapes in Solstice

18. Rib Shapes in Solstice • Solstice emulation uses circular arc ribs. • Plane determined by cross product of rib chord direction and normal of plane of minor circle.

19. Rib Shapes in Solstice • Solstice emulation uses circular arc ribs. • Plane determined by cross product of rib chord direction and normal of plane of minor circle.

20. Rib Shapes in Solstice • Solstice emulation uses circular arc ribs. • Plane determined by cross product of rib chord direction and normal of plane of minor circle.

21. Rib Shapes in Solstice

22. Solstice and Variations Solstice ( 3, 2 ) knot Modified ( 2, 3 ) knot

23. Solstice and Variations Solstice ( 3, 2 ) knot Modified ( 4, 3 ) knot

24. Solstice and Variations Solstice_2 ( 3, 2 ) knot (with denser ribs) Modified ( 4, 5 ) knot

25. Early Mace (Atlanta, GA) Emulation Variation with straight ribs Variation with convex ribs

26. Harmony (Hartford, CT) • Two semi-circular guide rails. • Four ribbed surfaces. • Ribs take off in direction of curve normal.

27. Ribbed Surfaces in Visualization Mathematician’s Models and Sculptures Boy’s Surface Hyperboloid String art by Ray Schechter

28. Ribbed Surfaces in Visualization Our Own Visualization Models Non-orientable, single-sided building blocks for the construction of abstract 4D polyhedra such as the 11-Cell and the 57-Cell.

29. Python Module • Python module for rapid development of design programs. • Quick and dirty creation of GUIthrough GLUI. • Supports output to RenderMan RIB format for high-quality rendering.

30. Conclusions • Ribbed surfaces are a concise representation of a broad range of sculptural forms: • Reduced weight and construction costs. • “Airy” realization, less shadows. Ribbed “transparency” ideal for visualization of self-intersecting surfaces. Naturally describes objects in architecture or in other design domains: • Balcony railings, furniture.

31. QUESTIONS?