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

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

Motivation  Inspired by Charles O. Perry's ribbed sculptures.  Emulate and generalize by abstraction.

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

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

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

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).

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) (C)(B)

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

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

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

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

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

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.

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

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.

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.

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.

Rib Shapes in Solstice

Solstice and Variations Modified ( 2, 3 ) knotSolstice ( 3, 2 ) knot

Solstice and Variations Modified ( 4, 3 ) knotSolstice ( 3, 2 ) knot

Solstice and Variations Modified ( 4, 5 ) knotSolstice_2 ( 3, 2 ) knot (with denser ribs)

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

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

Ribbed Surfaces in Visualization Mathematician’s Models and Sculptures HyperboloidBoy’s SurfaceString art by Ray Schechter

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.

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.

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.

QUESTIONS?