1. University of California, Berkeley
Florida 1999
M+D 2001, Geelong, July 2001
“Viae Globi”
Pathways on a Sphere
Carlo H. Séquin
University of California, Berkeley
 20 Minute Time slot !!

2. Computer-Aided Sculpture Design
Florida 1999
Computer-Aided Sculpture Design
I have been a CAD tool builder for last 20 years. I have actively become involved in sculpture design only in the last 6 years when I started a collaboration with Brent Collins.

3. “Hyperbolic Hexagon II” (wood)
Florida 1999
“Hyperbolic Hexagon II” (wood)
For whom I designed certain shapes on the computer which he then built oin 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
Florida 1999
Sculpture Generator 1 -- User Interface
Prototyping & Visualization tool for Scherk-Collins Saddle-Chains.
Slider control for this one shape-family,Control of about 12 parameters.
Main goal: Speed for interactive editing.

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

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

10. Profiled Slice through the Sculpture
Florida 1999
Profiled Slice through the Sculpture
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.
From these Collins will precut boards
then assemble the complete shape
and fine tune and polish it.

11. Another Joint Sculpture
Heptoroid

12. Inspiration: Brent Collins’ “Pax Mundi”

13. Keeping up with Brent ... Need a more general approach !
Sculpture Generator I can only do warped Scherk towers, not able to describe a shape like Pax Mundi.
Need a more general approach !
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.
Express it as a computer program
Insert parameters to change salient aspects / features of the sculpture
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
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
Can be seen as: Amplitude modulated, 4-period Gabo curve

18. SLIDE-UI for “Pax Mundi” Shapes
Florida 1999
SLIDE-UI for “Pax Mundi” Shapes
Good combination of interactive 3D graphics and parameterizable procedural constructs.
Good combination of interactive 3D graphics and parameterizable procedural constructs.

19. Advantages of CAD of Sculptures
Exploration of a larger domain
Instant visualization of results
Eliminate need for prototyping
Making more complex structures
Better optimization of chosen form
More precise implementation
Computer-generated output
Virtual reality displays
Rapid prototyping of maquettes
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)
periods

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

25. “Viae Globi 2”
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
Tried to maintain high degree of symmetry,
but wanted more highly convoluted paths …
Not as easy as I thought !
Tried to work with splines whose control vertices were placed at the vertices or edge mid-points of a Platonic or Archimedean polyhedron.
Tried to find Hamiltonian paths on the edges of a Platonic solid, but had only moderate success.
Used free-hand sketching on a sphere …

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

31. A Better CAD Tool is Needed !
A way to make nice curvy paths on the surface of a sphere: ==> C-splines.
A way to sweep interesting cross sections along these spherical paths: ==> SLIDE.
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?
Recursive scheme to create spline curves
using splitting and averaging
Example: Chaikin’s Algorithm
corner cutting algorithm ==> quadratic B-Spline
subdivision

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

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

36. Angle Division in the Plane
Find the point
that interpolates
the turning angles
at SL and SR
tS=(tL+ tR)/2

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

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

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 SL and SR, 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 … Need a plan !
Keep some symmetry !
Ideally high-order “spherical” symmetry.
Construct polyhedral path and smooth it.
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 Limited success with this formal approach:
either curve would not close
or it was one of the known configurations
Relax – just doodle with the editor …
Once a promising configuration had been found,
symmetrize the control points to the desired overall symmetry.
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
A rather winding Swiss mountain pass road in the upper Engadin.

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

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

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

50. Varying the Azimuth Parameter
Florida 1999
Varying the Azimuth Parameter
Setting the orientation of the cross section …
The shape of the sweep curve on the sphere is just ONE aspect of the sculpture !
We can also play with the cross section
And with the orientation of the cross section around the sweep curve.
… by Frenet frame
… using torsion-minimization with
two different azimuth values

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

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

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

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

55. This is a highly recommended approach in many engineering disciplines.
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…
QUESTIONS ?