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 Computer-Aided Sculpture DesignComputer-Aided Sculpture Design

CHSCHSUCBUCB ““Hyperbolic Hexagon II” (wood)Hyperbolic Hexagon II” (wood)

Brent Collins

CHSCHSUCBUCB Brent Collins: Stacked SaddlesBrent Collins: Stacked Saddles

CHSCHSUCBUCB Scherk’s 2nd Minimal SurfaceScherk’s 2nd Minimal Surface

Normal“biped”saddles

Generalization to higher-order saddles(monkey saddle)

CHSCHSUCBUCB Closing the LoopClosing the Loop

straight

or

twisted

CHSCHSUCBUCB Sculpture Generator 1 -- User InterfaceSculpture Generator 1 -- User Interface

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 Another Joint SculptureAnother Joint Sculpture

Heptoroid

CHSCHSUCBUCB

Inspiration: Inspiration:

Brent Collins’ Brent Collins’

“Pax Mundi”“Pax Mundi”

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 3-period Gabo curve3-period Gabo curve

Same construction as for as for 2-period curve

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 Fused Deposition Modeling (FDM)Fused Deposition Modeling (FDM)

CHSCHSUCBUCB Zooming into the FDM MachineZooming into the FDM Machine

CHSCHSUCBUCB FDM Part with SupportFDM Part with Support

as it comes out of the machine

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 Via Globi 3 (Stone)Via Globi 3 (Stone)

Wilmin Martono

CHSCHSUCBUCB Via Globi 5 (Wood)Via Globi 5 (Wood)

Wilmin Martono

CHSCHSUCBUCB Via Globi 5 (Gold)Via Globi 5 (Gold)

Wilmin Martono

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 Spherical C-SplinesSpherical C-Splines

use similar construction as in planar case

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 Via Globi -- Virtual DesignVia Globi -- Virtual Design

Wilmin Martono

CHSCHSUCBUCB ““Maloja” -- FDM partMaloja” -- FDM part

A rather winding Swiss mountain pass road in the upper Engadin.

CHSCHSUCBUCB ““Stelvio”Stelvio”

An even more convoluted alpine pass in Italy.

CHSCHSUCBUCB ““Altamont”Altamont”

Celebrating American multi-lane highways.

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.

CHSCHSUCBUCB The End of the Road…The End of the Road…