M+D 2001, Geelong, July 2001 “Viae Globi” Pathways on a Sphere Carlo H. Séquin University of California, Berkeley

Computer-Aided Sculpture Design

“Hyperbolic Hexagon II” (wood) Brent Collins

Brent Collins: Stacked Saddles

Scherk’s 2nd Minimal Surface Normal “biped” saddles Generalization to higher-order saddles (monkey saddle)

Closing the Loop straight or twisted

Sculpture Generator 1 -- User Interface

Brent Collins’ Prototyping Process Armature for the "Hyperbolic Heptagon" Mockup for the "Saddle Trefoil" Time-consuming ! (1-3 weeks)

Collins’ Fabrication Process Example: “Vox Solis” Layered laminated main shape Wood master pattern for sculpture

Profiled Slice through the Sculpture u One thick slice thru “Heptoroid” from which Brent can cut boards and assemble a rough shape. Traces represent: top and bottom, as well as cuts at 1/4, 1/2, 3/4 of one board.

Another Joint Sculpture u Heptoroid

Inspiration: Brent Collins’ “Pax Mundi”

Keeping up with Brent... u Sculpture Generator I can only do warped Scherk towers, not able to describe a shape like Pax Mundi. u Need a more general approach ! u Use the SLIDE modeling environment (developed at U.C. Berkeley by J. Smith) to capture the paradigm of such a sculpture in a procedural form. l Express it as a computer program l Insert parameters to change salient aspects / features of the sculpture l First: Need to understand what is going on     

Sculptures by Naum Gabo Pathway on a sphere: Edge of surface is like seam of tennis ball; ==> 2-period Gabo curve.

2-period Gabo curve u Approximation with quartic B-spline with 8 control points per period, but only 3 DOF are used.

3-period Gabo curve Same construction as for as for 2-period curve

“Pax Mundi” Revisited u Can be seen as: Amplitude modulated, 4-period Gabo curve

SLIDE-UI for “Pax Mundi” Shapes Good combination of interactive 3D graphics and parameterizable procedural constructs.

Advantages of CAD of Sculptures u Exploration of a larger domain l Instant visualization of results l Eliminate need for prototyping u Making more complex structures l Better optimization of chosen form l More precise implementation u Computer-generated output l Virtual reality displays l Rapid prototyping of maquettes l Milling of large-scale master for casting

Fused Deposition Modeling (FDM)

Zooming into the FDM Machine

FDM Part with Support as it comes out of the machine

“Viae Globi” Family (Roads on a Sphere) periods

2-period Gabo sculpture u Looks more like a surface than a ribbon on a sphere.

“Viae Globi 2” u Extra path over the pole to fill sphere surface more completely.

Via Globi 3 (Stone) Wilmin Martono

Via Globi 5 (Wood) Wilmin Martono

Via Globi 5 (Gold) Wilmin Martono

Towards More Complex Pathways u Tried to maintain high degree of symmetry, u but wanted more highly convoluted paths … u Not as easy as I thought ! u Tried to work with splines whose control vertices were placed at the vertices or edge mid-points of a Platonic or Archimedean polyhedron. u Tried to find Hamiltonian paths on the edges of a Platonic solid, but had only moderate success. u Used free-hand sketching on a sphere …

Conceiving “Viae Globi” u Sometimes I started by sketching on a tennis ball !

A Better CAD Tool is Needed ! u A way to make nice curvy paths on the surface of a sphere: ==> C-splines. u A way to sweep interesting cross sections along these spherical paths: ==> SLIDE. u A way to fabricate the resulting designs: ==> Our FDM machine.

Circle-Spline Subdivision Curves Carlo Séquin Jane Yen on the plane -- and on the sphere

Review: What is Subdivision? u Recursive scheme to create spline curves l using splitting and averaging u Example: Chaikin’s Algorithm l corner cutting algorithm ==> quadratic B-Spline subdivision

An Interpolating Subdivision Curve u 4-point cubic interpolation in the plane: S = 9B/16 + 9C/16 – A/16 – D/16 A B D CM S

Interpolation with Circles u Circle through 4 points – if we are lucky … u If not: left circle ; right circle ; interpolate. A B D C S The real issue is how this interpolation should be performed ! SLSL SRSR

Angle Division in the Plane Find the point that interpolates the turning angles at S L and S R  S =(  L +  R )/2

C-Splines u Interpolate constraint points. u Produce nice, rounded shapes. u Approximate the Minimum Variation Curve (MVC) l minimizes squared magnitude of derivative of curvature l fair, “natural”, “organic” shapes u Geometric construction using circles: l not affine invariant - curves do not transforms exactly as their control points (except for uniform scaling). l Advantages: can produce circles, avoids overshoots l Disadvantages: n cannot use a simple linear interpolating mask / matrix n difficult to analyze continuity, etc

Various Interpolation Schemes The new C-Spline Classical Cubic Interpolation Linearly Blended Circle Scheme Too “loopy” 1 step 5 steps

Spherical C-Splines  use similar construction as in planar case

Seamless Transition: Plane - Sphere In the plane we find S by halving an angle and intersecting with line m. On the sphere we originally wanted to find S L and S R, and then find S by halving the angle between them. ==> Problems when BC << sphere radius. Do angle-bisection on an outer sphere offset by d/2.

Circle Splines on the Sphere Examples from Jane Yen’s Editor Program

Now We Can Play … ! But not just free-hand drawing … u Need a plan ! u Keep some symmetry ! u Ideally high-order “spherical” symmetry. u Construct polyhedral path and smooth it. u Start with Platonic / Archemedean solids.

Hamiltonian Paths Strictly realizable only on octahedron!  Gabo-2 path. Pseudo Hamiltonian path (multiple vertex visits)  Gabo-3 path.

Other Approaches u Limited success with this formal approach: l either curve would not close l or it was one of the known configurations u Relax – just doodle with the editor … Once a promising configuration had been found, l symmetrize the control points to the desired overall symmetry. l fine-tune their positions to produce satisfactory coverage of the sphere surface. Leads to nice results … 

Via Globi -- Virtual Design Wilmin Martono

“Maloja” -- FDM part u A rather winding Swiss mountain pass road in the upper Engadin.

“Stelvio” u An even more convoluted alpine pass in Italy.

“Altamont” u Celebrating American multi-lane highways.

“Lombard” u A very famous crooked street in San Francisco u Note that I switched to a flat ribbon.

Varying the Azimuth Parameter Setting the orientation of the cross section … … by Frenet frame … using torsion-minimization with two different azimuth values

“Aurora” u Path ~ Via Globi 2 u Ribbon now lies perpendicular to sphere surface. u Reminded me of the bands in an Aurora Borrealis.

“Aurora - T” u Same sweep path ~ Via Globi 2 u Ribbon now lies tangential to sphere surface.

“Aurora – F” (views from 3 sides) u Still the same sweep path ~ Via Globi 2 u Ribbon orientation now determined by Frenet frame.

“Aurora-M” u Same path on sphere, u but more play with the swept cross section. u This is a Moebius band. u It is morphed from a concave shape at the bottom to a flat ribbon at the top of the flower.

Conclusions An example where a conceptual design-task, mathematical analysis, and tool-building go hand-in-hand. This is a highly recommended approach in many engineering disciplines.

The End of the Road…