CS285CS285Designing Viae Globi

(Roads on a Sphere)

Carlo H. Séquin

University of California, Berkeley

Inspired by

Brent Collins

Gower, Missouri

““Hyperbolic Hexagon” by B. CollinsHyperbolic Hexagon” by B. Collins

6 saddles in a ring

6 holes passing through symmetry plane at ±45º

= “wound up” 6-story Scherk tower

Discussion: What if … we added more stories ?

or introduced a twist before closing the ring ?

Closing the LoopClosing the Loop

straight

or

twisted

““Sculpture Generator I”, GUI Sculpture Generator I”, GUI

Base Geometry: One Scherk StoryBase Geometry: One Scherk Story

Hyperbolic Slices ==> Triangle Strips

Pre-computed -- then warped into toroid

Brent CollinsBrent Collins

“Hyperbolic Hexagon II”

Hyper-Sculpture: Hyper-Sculpture: Family of 12 TrefoilsFamily of 12 Trefoils

W=2

W=1

B=1 B=2 B=3 B=4

Brent Collins’ Brent Collins’ Pax MundiPax Mundi

1997: wood, 30”diam.1997: wood, 30”diam.

2006: Commission from H&R Block, Kansas Cityto make a 70”diameter version in bronze.

My task: to define the master geometry.

CAD tools played important role.

How to Model How to Model Pax MundiPax Mundi ... ...

Already addressed that question in 1998:

Pax Mundi could not be done with Sculpture Generator I

Needed a more general program !

Used the Berkeley SLIDE environment.

First: Needed to find the basic paradigm

Sculptures by Naum GaboSculptures by Naum Gabo

Pathway on a sphere:

Edge of surface is like seam of tennis- or base-ball;

2-period Gabo curve.

2-period “Gabo Curve”2-period “Gabo Curve”

Approximation with quartic B-splinewith 8 control points per period,but only 3 DOF are used (symmetry!).

4-period “Gabo Curve”4-period “Gabo Curve”

Same construction as for as for 2-period curve

Pax MundiPax Mundi Revisited Revisited

Can be seen as:

Amplitude modulated, 4-period Gabo curve

SLIDESLIDE

SLIDE = Scene Language for Interactive Dynamic Environments

Developed as a modular rendering pipelinefor our introductory graphics course.

Primary Author: Jordan Smith

Based on OpenGL and Tcl/tk.

Good combination of interactive 3D graphicsand parameterizable procedural constructs.

SLIDE Example: Klein BottleSLIDE Example: Klein Bottle

Final Project CS 184, Nerius Landys & Shad Roundy

SLIDE Example Bug’s LifeSLIDE Example Bug’s Life

Final Project CS 184, David Cheng and James Chow

SLIDE as a Design ToolSLIDE as a Design Tool

SLIDE originally a modular rendering tool.

Later enhanced to serve as a CAD tool:

Spline curves and surfaces

Morphing sweeps along such curves

3D warping module (Sederberg, Rockwood)

Many types of subdivision surfaces

These are key elements for a 2nd Generation Sculpture Generator

SLIDE-UI for Knot GenerationSLIDE-UI for Knot Generation

SLIDE-GUI for “SLIDE-GUI for “Pax MundiPax Mundi” Shapes” ShapesGood combination of interactive 3D graphicsand parameterizable procedural constructs.

2-period Gabo sculpture2-period Gabo sculpture

Tennis ball – or baseball – seam used assweep curve.

Viae Globi Viae Globi Family Family (Roads on a Sphere)(Roads on a Sphere)

2 3 4 5 periods

Via Globi 5Via Globi 5 (Virtual Wood) (Virtual Wood)

Wilmin Martono

Modularity of Modularity of Gabo Curve GeneratorGabo Curve Generator

Sweep Curve Generator:

Gabo Curves as B-splines

Cross Section Fine Tuner:

Paramererized shapes

Sweep / Twist Controller

Sweep / Twist ControlSweep / Twist Control

How do we orient, move, morph ...the cross section along the sweep path ?

Natural orientationwith Frenet frame

Torsion Minimization:Azimuth: tangential / normal

900° of twistadded.

Extension:Extension: Free-form Curve on a Sphere Free-form Curve on a Sphere

Spherical Spline Path Editor (Jane Yen)

Nice smooth interpolating curves through sparse data points

Many Different Many Different Viae GlobiViae Globi Models Models

MalojaMaloja

Extending the Paradigm:Extending the Paradigm: Aurora-MAurora-M

Simple 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.

Paradigm Extension:Paradigm Extension: Sweep Path Sweep Path is no longer confined to a sphere!is no longer confined to a sphere!

Music of the Spheres (Brent Collins)

Allows Knotted Sweep PathsAllows Knotted Sweep Paths

Chinese Button Knot

Really Free-form 3D Space CurvesReally Free-form 3D Space Curves

Figure-8 knot

Using Another Cross SectionUsing Another Cross Section

Galapagos 6

Target GeometryTarget Geometry

Constraints:• Bronze, 70” diameter• Less than 1500 pounds• Less than $50’000• Maintain beauty, strength• Minimize master geometry

EmulationEmulation

Use smoother quintic spline

Make 4 identical parts(needs extra control point)

Add a few “warp” parameters

Emulation; Define Master PatternEmulation; Define Master Pattern

Use 4 copies.

Master to make a mold from.

Alignment tab

Model of Master Part Made with FDMModel of Master Part Made with FDM

4 pieces make the whole sculpture

Joe Valasek’s CNC Milling MachineJoe Valasek’s CNC Milling Machine

Styrofoam milling machine

Design of Two-Part MasterDesign of Two-Part Master

Alignment tabs for easy assembly

Subdivide into Subdivide into TwoTwo Master Segments Master Segments

Machined Master Pattern #2Machined Master Pattern #2

(Cut) Master (Cut) Master Silicone Rubber Silicone Rubber MoldMold

MoldMold Several (4) Wax Copies Several (4) Wax Copies

SpruingSpruing the the Wax PartsWax Parts for Casting for Casting

Ceramic Slurry ShellCeramic Slurry Shell Around Wax Around Wax PartPart

ShellShell Ready Ready for Castingfor Casting

Casting with Casting with Liquid BronzeLiquid Bronze

The Freed The Freed Bronze CastBronze Cast

Assembling the SegmentsAssembling the Segments

The “Growing” RibbonThe “Growing” Ribbon

The Single Support PointThe Single Support Point

Grinding the Welded Seams,Grinding the Welded Seams,Polishing the SurfacePolishing the Surface

Applying PatinaApplying Patina

Ready for ShippingReady for Shipping

The Move Around the Building (Jan.18)The Move Around the Building (Jan.18)

Front DoorFront Door

H&R Block Building

Steve Tightening the BoltsSteve Tightening the Bolts

Brent Polishing Our BabyBrent Polishing Our Baby