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

2. Computer-Aided Sculpture Design

3. “Hyperbolic Hexagon II” (wood) Brent Collins

4. Brent Collins: Stacked Saddles

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

6. Closing the Loop straight or twisted

7. Sculpture Generator 1 -- User Interface

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

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

10. 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.

11. Another Joint Sculpture u Heptoroid

12. Inspiration: Brent Collins’ “Pax Mundi”

13. 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     

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

15. 2-period Gabo curve u Approximation with quartic B-spline with 8 control points per period, but only 3 DOF are used.

16. 3-period Gabo curve Same construction as for as for 2-period curve

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

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

19. 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

20. Fused Deposition Modeling (FDM)

21. Zooming into the FDM Machine

22. FDM Part with Support as it comes out of the machine

23. “Viae Globi” Family (Roads on a Sphere) 2 3 4 5 periods

24. 2-period Gabo sculpture u Looks more like a surface than a ribbon on a sphere.

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

26. Via Globi 3 (Stone) Wilmin Martono

27. Via Globi 5 (Wood) Wilmin Martono

28. Via Globi 5 (Gold) Wilmin Martono

29. 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 …

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

31. 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.

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

33. 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

34. 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

35. 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

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

37. 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

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

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

40. 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.

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

42. 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.

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

44. 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 … 

45. Via Globi -- Virtual Design Wilmin Martono

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

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

48. “Altamont” u Celebrating American multi-lane highways.

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

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

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

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

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

54. “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.

55. 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.

56. The End of the Road…