M+D 2001, Geelong, July 2001 “Viae Globi” Pathways on a Sphere Carlo H. Séquin University of California, Berkeley
“Hyperbolic Hexagon II” (wood) Brent Collins
Scherk’s 2nd Minimal Surface Normal “biped” saddles Generalization to higher-order saddles(monkey saddle)
Closing the Loop straight or twisted
Brent Collins’ Prototyping Process Mockup for the "Saddle Trefoil" Armature for the "Hyperbolic Heptagon" Time-consuming ! (1-3 weeks)
Collins’ Fabrication Process Wood master patternfor sculpture Layered laminated main shape Example: “Vox Solis”
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.
Another Joint Sculpture • Heptoroid
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
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 • Approximation with quartic B-splinewith 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 • Can be seen as:Amplitude modulated, 4-period Gabo curve
SLIDE-UI for “Pax Mundi” Shapes Good combination of interactive 3D graphicsand parameterizable procedural constructs.
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
FDM Part with Support as it comes out of the machine
“Viae Globi” Family (Roads on a Sphere) 2 3 4 5 periods
2-period Gabo sculpture • Looks more like a surface than a ribbon on a sphere.
“Viae Globi 2” • Extra path over the poleto 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 • 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 …
Conceiving “Viae Globi” • Sometimes I started by sketching on a tennis ball !
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.
Circle-Spline Subdivision Curves Carlo Séquin Jane Yen on the plane -- and on the sphere
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
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
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 !
Angle Division in the Plane Find the point that interpolates the turning angles at SL and SR tS=(tL+ tR)/2
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
Various Interpolation Schemes 1 step 5 steps Too “loopy” ClassicalCubicInterpolation LinearlyBlendedCircle Scheme The new C-Spline
Spherical C-Splines use similar construction as in planar case
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.
Circle Splines on the Sphere Examples from Jane Yen’s Editor Program
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.
Hamiltonian Paths Strictly realizable only on octahedron! Gabo-2 path. Pseudo Hamiltonian path (multiple vertex visits) Gabo-3 path.
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 …
Via Globi -- Virtual Design Wilmin Martono
“Maloja” -- FDM part • A rather winding Swiss mountain pass road in the upper Engadin.
“Stelvio” • An even more convoluted alpine pass in Italy.
“Altamont” • Celebrating American multi-lane highways.
“Lombard” • A very famous crooked street in San Francisco • Note that I switched to a flat ribbon.
Varying the Azimuth Parameter Setting the orientation of the cross section … … using torsion-minimization with two different azimuth values … by Frenet frame