CS285
Designing Viae Globi
(Roads on a Sphere)
Carlo H. Séquin
University of California, Berkeley
Inspired by
Brent Collins
Gower, Missouri
“Hyperbolic 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 Loop
straight
or
twisted
“Sculpture Generator I”, GUI
Base Geometry: One Scherk Story
 Hyperbolic
Slices ==> Triangle Strips
 Pre-computed
-- then warped into toroid
Brent Collins
“Hyperbolic Hexagon II”
Hyper-Sculpture: Family of 12 Trefoils
W=2
W=1
B=1
B=2
B=3
B=4
Brent Collins’
Pax Mundi
1997: wood, 30”diam.
2006: Commission from
H&R Block, Kansas City
to make a 70”diameter
version in bronze.
My task:
to define the master geometry.
CAD tools played important role.
How to Model Pax 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 Gabo
Pathway on a sphere:
Edge of surface is like seam of tennis- or base-ball;
 2-period Gabo curve.
2-period “Gabo Curve”
 Approximation
with quartic B-spline
with 8 control points per period,
but only 3 DOF are used (symmetry!).
4-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
SLIDE = Scene Language for
Interactive Dynamic Environments
Developed as a modular rendering pipeline
for our introductory graphics course.
Primary Author: Jordan Smith

Based on OpenGL and Tcl/tk.

Good combination of interactive 3D graphics
and parameterizable procedural constructs.
SLIDE Example: Klein Bottle
Final Project CS 184, Nerius Landys & Shad Roundy
SLIDE Example Bug’s Life
Final Project CS 184, David Cheng and James Chow
SLIDE as a Design Tool
 SLIDE
 Later
originally a modular rendering tool.
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 Generation
SLIDE-GUI for “Pax Mundi” Shapes
Good combination of interactive 3D graphics
and parameterizable procedural constructs.
2-period Gabo sculpture
Tennis ball –
or baseball –
seam
used as
sweep curve.
Viae Globi Family
2
3
(Roads on a Sphere)
4
5
periods
Via Globi 5 (Virtual Wood)
Wilmin Martono
Modularity of Gabo Curve Generator
 Sweep

Gabo Curves as B-splines
 Cross

Curve Generator:
Section Fine Tuner:
Paramererized shapes
 Sweep
/ Twist Controller
Sweep / Twist Control

How do we orient, move, morph ...
the cross section along the sweep path ?
Natural orientation
with Frenet frame
Torsion Minimization:
Azimuth: tangential / normal
900° of twist
added.
Extension: Free-form Curve on a Sphere
Spherical Spline Path Editor (Jane Yen)
Nice smooth interpolating curves through sparse data points
Many Different Viae Globi Models
Maloja
Extending the Paradigm: Aurora-M
 Simple
path on sphere,
 but
more play with the
swept cross section.
 This
 It
is a Moebius band.
is morphed from a
concave shape at the
bottom to a flat ribbon
at the top of the flower.
Paradigm Extension: Sweep Path
is no longer confined to a sphere!
Music of the Spheres (Brent Collins)
Allows Knotted Sweep Paths
Chinese Button Knot
Really Free-form 3D Space Curves
Figure-8 knot
Using Another Cross Section
Galapagos 6
Target Geometry
Constraints:
• Bronze, 70” diameter
• Less than 1500 pounds
• Less than $50’000
• Maintain beauty, strength
• Minimize master geometry
Emulation

Use smoother quintic spline

Make 4 identical parts
(needs extra control point)

Add a few “warp” parameters
Emulation; Define Master Pattern
 Master
to make
a mold from.
Alignment tab

Use 4 copies.
Model of Master Part Made with FDM

4 pieces make the whole sculpture
Joe Valasek’s CNC Milling Machine
 Styrofoam
milling
machine
Design of Two-Part Master
 Alignment
tabs for easy assembly
Subdivide into Two Master Segments
Machined Master Pattern #2
(Cut) Master  Silicone Rubber
Mold
Mold  Several (4) Wax Copies
Spruing the Wax Parts for Casting
Ceramic Slurry Shell Around Wax
Part
Shell Ready
for Casting
Casting with Liquid Bronze
The Freed Bronze Cast
Assembling the Segments
The “Growing” Ribbon
The Single Support Point
Grinding the Welded Seams,
Polishing the Surface
Applying Patina
Ready for Shipping
The Move Around the Building (Jan.18)
Front Door
H&R Block Building
Steve Tightening the Bolts
Brent Polishing Our Baby