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M+D 2001, Geelong, July 2001

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.

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M+D 2001, Geelong, July 2001

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  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 Mockup for the "Saddle Trefoil" Armature for the "Hyperbolic Heptagon" Time-consuming ! (1-3 weeks)

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

  10. Profiled Slice through the Sculpture • One thick slicethru “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, 3/4of one board.

  11. Another Joint Sculpture • Heptoroid

  12. Inspiration: Brent Collins’ “Pax Mundi”

  13. Keeping up with Brent ... • Sculpture Generator Ican 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 sculpturein 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-splinewith 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 Good combination of interactive 3D graphicsand 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) 2 3 4 5 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 poleto 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 pathson 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 sectionsalong 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. subdivision subdivision Review: What is Subdivision? • Recursive scheme to create spline curves • using splitting and averaging • Example: Chaikin’s Algorithm • corner cutting algorithm ==> quadratic B-Spline

  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” ClassicalCubicInterpolation LinearlyBlendedCircle 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 Sby halving an angle andintersecting with line m. On the sphere we originallywanted to find SL and SR,and then find S by halvingthe 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 Setting the orientation of the cross section … … using torsion-minimization with two different azimuth values … by Frenet frame

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