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CHSCHSUCBUCB M+D 2001, Geelong, July 2001M+D 2001, Geelong, July 2001
“Viae Globi”
Pathways on a Sphere
Carlo H. Séquin
University of California, Berkeley
CHSCHSUCBUCB Scherk’s 2nd Minimal SurfaceScherk’s 2nd Minimal Surface
Normal“biped”saddles
Generalization to higher-order saddles(monkey saddle)
CHSCHSUCBUCB Brent Collins’ Prototyping ProcessBrent Collins’ Prototyping Process
Armature for the "Hyperbolic Heptagon"
Mockup for the "Saddle Trefoil"
Time-consuming ! (1-3 weeks)
CHSCHSUCBUCB Collins’ Fabrication ProcessCollins’ Fabrication Process
Example: “Vox Solis”
Layered laminated main shapeWood master pattern
for sculpture
CHSCHSUCBUCB Profiled Slice through the SculptureProfiled 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.
CHSCHSUCBUCB Keeping up with Brent ...Keeping up with Brent ...
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 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
CHSCHSUCBUCB Sculptures by Naum GaboSculptures by Naum Gabo
Pathway on a sphere:
Edge of surface is like seam of tennis ball;
==> 2-period Gabo curve.
CHSCHSUCBUCB 2-period Gabo curve2-period Gabo curve
Approximation with quartic B-splinewith 8 control points per period,but only 3 DOF are used.
CHSCHSUCBUCB ““Pax Mundi” RevisitedPax Mundi” Revisited
Can be seen as:
Amplitude modulated, 4-period Gabo curve
CHSCHSUCBUCB SLIDE-UI for “Pax Mundi” ShapesSLIDE-UI for “Pax Mundi” Shapes
Good combination of interactive 3D graphicsand parameterizable procedural constructs.
CHSCHSUCBUCB Advantages of CAD of SculpturesAdvantages 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
CHSCHSUCBUCB ““Viae Globi” Family Viae Globi” Family (Roads on a Sphere)(Roads on a Sphere)
2 3 4 5 periods
CHSCHSUCBUCB 2-period Gabo sculpture2-period Gabo sculpture
Looks more like a surface than a ribbon on a sphere.
CHSCHSUCBUCB ““Viae Globi 2”Viae Globi 2”
Extra path over the poleto fill sphere surface more completely.
CHSCHSUCBUCB Towards More Complex PathwaysTowards 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 …
CHSCHSUCBUCB Conceiving “Viae Globi”Conceiving “Viae Globi”
Sometimes I started by sketching on a tennis ball !
CHSCHSUCBUCB A Better CAD Tool is Needed !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.
CHSCHSUCBUCB Circle-Spline Subdivision CurvesCircle-Spline Subdivision Curves
Carlo SéquinJane Yen
on the plane -- and on the sphere
CHSCHSUCBUCB Review: What is 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
subdivisionsubdivision
CHSCHSUCBUCB An Interpolating Subdivision CurveAn Interpolating Subdivision Curve
4-point cubic interpolation in the plane:
S = 9B/16 + 9C/16 – A/16 – D/16
A
B
D
CM
S
CHSCHSUCBUCB Interpolation with CirclesInterpolation with Circles
Circle through 4 points – if we are lucky …
If not: left circle ; right circle ; interpolate.
A
B
D
C
S
The real issue is how this interpolation should be performed !
SL
SR
CHSCHSUCBUCB Angle Division in the PlaneAngle Division in the Plane
Find the point
that interpolates
the turning angles
at SL and SR
S=(L+ R)/2
CHSCHSUCBUCB C-SplinesC-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
dsdsd 2)(
CHSCHSUCBUCB Various Interpolation SchemesVarious Interpolation Schemes
The new C-Spline
ClassicalCubic
Interpolation
LinearlyBlended
Circle Scheme
Too “loopy”
1 step
5 steps
CHSCHSUCBUCB Seamless Transition: Plane - SphereSeamless 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.
CHSCHSUCBUCB Circle Splines on the SphereCircle Splines on the Sphere
Examples from Jane Yen’s Editor Program
CHSCHSUCBUCB Now We Can Play … !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.
CHSCHSUCBUCB Hamiltonian PathsHamiltonian Paths
Strictly realizable only on octahedron! Gabo-2 path.
Pseudo Hamiltonian path (multiple vertex visits) Gabo-3 path.
CHSCHSUCBUCB Other ApproachesOther 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 …
CHSCHSUCBUCB ““Maloja” -- FDM partMaloja” -- FDM part
A rather winding Swiss mountain pass road in the upper Engadin.
CHSCHSUCBUCB ““Lombard”Lombard”
A very famous crooked street in San Francisco
Note that I switched to a flat ribbon.
CHSCHSUCBUCB Varying the Azimuth ParameterVarying the Azimuth Parameter
Setting the orientation of the cross section …
… by Frenet frame … using torsion-minimization withtwo different azimuth values
CHSCHSUCBUCB ““Aurora”Aurora”
Path ~ Via Globi 2
Ribbon now lies perpendicular to sphere surface.
Reminded me ofthe bands in anAurora Borrealis.
CHSCHSUCBUCB ““Aurora - T”Aurora - T”
Same sweep path ~ Via Globi 2
Ribbon now lies tangential to sphere surface.
CHSCHSUCBUCB ““Aurora – F” (views from 3 sides)Aurora – F” (views from 3 sides)
Still the same sweep path ~ Via Globi 2
Ribbon orientation now determined by Frenet frame.
CHSCHSUCBUCB ““Aurora-M”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.
CHSCHSUCBUCB ConclusionsConclusions
An example where a conceptual design-task,
mathematical analysis,and tool-building go hand-in-hand.
This is a highly recommended approachin many engineering disciplines.