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

2. Overview Charles O. Perry's Solstice Reverse engineering Solstice
Ribbed Surface Paradigm
Solstice Program
Visualization of complex surfaces
Ribbed surfaces in architecture
In this presentation, I’ll begin by presenting sculptor Charles Perry’s sculpture Solstice,
Installed in Tampa Bay, FL. Solstice has served as the motivator for the development of
Ribbed surfaces. They emerged from the reverse engineering of this and other of Perry’s
Works in this so-called ribbed style. After describing in general terms the reverse
Engineering of Solstice, I provide an overview of ribbed surfaces, their parameters. 2

3. Motivation Inspired by Charles O. Perry's ribbed sculptures.
Emulate and generalize by abstraction.
We began with trying to get an intimate understanding of Charles Perry’s ribbed sculptures, his sculpture Solstice in particular. We reverse engineer the sculpture, identifying variables that may be altered to produce variations on the work. One element of this abstraction, ribbed surfaces, survives beyond this emulation and generalization of a particular work and has suggested uses for other works and in other domains. 3

4. Solstice is installed in Tampa Bay, Fl
Solstice is installed in Tampa Bay, Fl. It is characterized by wildly varying views, dramatic lines, and an unimposing presence; the ribbed realization casts minimal shadows and allows the background to be seen. 4

5. Parameterization of Solstice
(3, 2) torus knot
Curved “ribs” in nearly
triangular configuration
Now I’d like to explain the reverse engineering of Solstice. This thicker tube, which I’ll be referring to as the “guide rail”, is in fact a (3, 2) torus knot. That is, it is a knot that hugs the surface of a torus, making three revolutions about the major circle of the torus, and two revolutions about the minor circle of the torus. As there are three revolutions around the major circle, through any given slice of the toroidal arm the guide knot passes three times. The ribs connect these points on the guide rail to form a triangle. Thinking about it like this, one can imagine revolving such a triangle around the center of the torus, rotating it 2/3. In the sculpture, the ribs only nearly form triangles. For one, they are in fact concave, bent towards the centers of the minor circles of the torus. 5

6. Parameterization of Solstice
(3, 2) torus knot
Curved “ribs” in nearly
triangular configuration
Second, they are in fact skewed slightly along the guide rail, so that they are actually staggered, one rib in one direction, the next in the other direction, and so on. Thus following the ribs one actually traces out a helix. 6

7. Parameterization of Solstice
Thus we have identified three parameters: the type of torus knot, the staggering of ribs along the guide rail, and the concavity of the ribs. Also, trivially, the number of ribs may be changed. Each of these parameters may be altered independently to produce a unique sculpture.
Guide rail:
(3, 2) torus knot
Staggering of ribs: rib offset along guide rail
Rib shapes:
concave “hyperbolic” triangles

8. Ribbed Surfaces Guide rail(s) Ribs swept along 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).
From this we come to ribbed surfaces as an element for designing such works. Ribbed surfaces are specified in terms of the guide rails, which depend entirely on the application. In the generalization of Solstice, we have a torus knot as a single guide rail.
The ribs are then swept along the guide rail(s). The shapes of the ribs are determined procedurally, often in terms of guide rail derivative information.
In applications for which ribbed surfaces are suitable, the abstraction presented here reduces the number of parameters that must be specified to generate the surface. Compared to, for example, the sweep surfaces in Maya…

9. Sweep Surfaces One or two path or rail curves
One or two more profile curves
Maya: Extrusions (A), Lofts (B), Bi-Rails (C).
(A)
Compared to, for example, the sweep surfaces in Maya, we specify the guide rails and some parameters to the rib generator, on the order of a constant number. Moreover, the. Extrusions don’t suffice. Lofts require the specification of N profile curves. Birails come close, but like lofts require the specification of N profile curves.
(B)
(C) 9

10. Sweeping Ribs Single rail [0, 0.5) → [0.5, 1.0) Two rails
Ribs are then swept along the parameterized guide rail(s). They form a “mapping” between intervals on the guide rails. Here we show both the single rail case and the dual rail case. For a single rail, we can map between different or offset intervals to produce the surface. Here, we have a ruled Moebius strip. The single rail case can actually be thought of as a special case of the two rail situation. Think of it as the two-rail case where the two rails happen to be identical.
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 )
Guide rails are application-specific, and form the base of the form. Again, in Solstice, (p, q) torus knots are used.
( 2, 3 )
( 3, 2 )

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

13. Rib Parameterization θ Cubic Hermite Circular Arcs
Tangent directions and magnitudes at both ends
Circular Arcs
Embedding plane
Turning angle
Rails
Rails θ
Finally, we arrive at rib shape. We have so far found use for two representations for ribs. One, as cubic hermite splines, where the endpoints are determined by the rail and the tangent directions and magnitudes are determined procedurally. The second representation, and one that lends itself to manufacture, is as circular arcs. Here we need to specify an embedding plane and the turning angle.

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

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 d from 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 25

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

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 GUI through 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. 30

31. QUESTIONS? 31