Southern Oregon University, May 2003

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Southern Oregon University, May 2003

• Surface Optimization and Aesthetic Engineering

Carlo Séquin, University of California,
Berkeley
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I am a Designer
CCD Camera, Bell Labs, 1973 Soda Hall,
Berkeley, 1994
RISC chip, Berkeley, 1981 Octa-Gear,
Berkeley, 2000
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Focus of Talk

• The role of the computer in
• the creative process,
• aesthetic optimization.

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Outline

• Collaboration with Brent Collins
• Parameterized Shape Generation
• Realization by Layered Manufacturing
• Geometric Sculptures in Snow
• Aesthetics of Minimal Surfaces
• Sphere Inversion as a Challenge
• Search for a Beauty Functional
• CAD Tools that We Are Lacking

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Leonardo -- Special Issue
On Knot-Spanning Surfaces An Illustrated Essay
on Topological Art With an Artists Statement by
Brent Collins
George K. Francis with Brent Collins
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Brent Collins
Hyperbolic Hexagon II
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Scherks 2nd Minimal Surface
Normal biped saddles
Generalization to higher-order saddles(monkey
saddle)
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Brent Collins Stacked Saddles
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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 ?

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Closing the Loop
straight or twisted
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Brent Collins Prototyping Process
Mockup for the "Saddle Trefoil"
Armature for the "Hyperbolic Heptagon"
Time-consuming ! (1-3 weeks)
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Sculpture Generator I, GUI
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A Simple Scherk-Collins Toroid

• Parameters(genome)
• branches 2
• stories 1
• height 5.00
• flange 1.00
• thickness 0.10
• rim_bulge 1.00
• warp 360.00
• twist 90
• azimuth 90
• textr_tiles 3
• detail 8

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A Scherk Tower (on its side)

• branches 7
• stories 3
• height 0.2
• flange 1.00
• thickness 0.04
• rim_bulge 0
• warp 0
• twist 0
• azimuth 0
• textr_tiles 2
• detail 6

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A Virtual Sculpture (1996)
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V-art
VirtualGlassScherkTowerwith MonkeySaddles(R
adiance 40 hours) Jane Yen
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Minimal Surfaces
Catenoid

• At all surface points, Minimal Surfaceshave
  equal and opposite principal curvatures.

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Main Goal in Sculpture Generator 1

• Real-time Interactive Speed !
• Cant afford real surface optimizationto obtain
  true minimal surfaces (too slow)
• also, this would be aesthetically too limited.
• ? Make closed-form hyperbolic approximation.

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Hyperbolic Cross Sections
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Base Geometry One Scherk Story

• Hyperbolic Slices ? Triangle Strips
• precomputed ? then warped into toroid

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The Basic Saddle Element

• with surface normals

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Hyperbolic Contour Lines

• On a straight tower and on a toroidal ring

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Part IIIHow to Obtain a Real Sculpture ?

• Prepare a set of cross-sectional blue printsat
  equally spaced height intervals,corresponding
  to the board thickness that Collins is using
  for the construction.

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Collins Fabrication Process
Wood master patternfor sculpture
Layered laminated main shape
Example Vox Solis
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Emergence of the Heptoroid (1)
Assembly of the precut boards
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Emergence of the Heptoroid (3)
Smoothing the whole surface
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Slices through Minimal Trefoil
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10
23
30
45
5
20
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2
15
25
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SFF (Solid Free-form Fabrication)
Monkey- Saddle Cinquefoil
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Fused Deposition Modeling (FDM)
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Zooming into the FDM Machine
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Various Scherk-Collins Sculptures
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Part IV

• But what, if we want to make a really large
  sculpture ?

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Breckenridge, 2003

• Brent Collins and Carlo Séquin
• are invited to join the team
• and to provide a design.
• Other Team Members
• Stan Wagon, Dan Schwalbe, Steve Reinmuth
• ( Team Minnesota)

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Stan Wagon, Macalester College, St. Paul, MN

• Leader of Team USA Minnesota

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Breckenridge, 1999

• Helaman Ferguson Invisible Handshake

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Breckenridge, 2000

• Robert Longhurst
• Rhapsody in White
• 2nd Place

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Monkey Saddle Trefoil

• from Sculpture Generator I

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The Poor Mans Opportunity Snow-Sculpting!Annua
l Championships in Breckenridge, CO
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Whirled White Web
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(No Transcript)
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1240 pm -- 42 F
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124001
Photo StRomain
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1241 pm -- 42 F
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The Winners

• 1st Canada B.C., 2nd USA
  Minnesota, 3rd USA Breckenridge

sacred geometry very intricate very 21st
century !
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4 pm
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Snow Sculpting

• More on the construction and drama of our snow
  sculpture tonight at 7pm.
• Also, pictures of some of the other snow
  sculptures.

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Part V

• DISCUSSION
• Aesthetics of Minimal Surfaces

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Whirled White Web Séquin 2003
Minimal surface spanning three (2,1) torus knots
Maquette made with Sculpture Generator I
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Tightest Saddle Trefoil Séquin 1997
Shape generated with Sculpture Generator 1
Minimal surface spanning one (4,3) torus knots
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Atomic Flower II by Brent Collins

• Minimal surface in smooth edge(captured by John
  Sullivan)

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Surface by P. J. Stewart (J. Hrdlicka)

• Minimal surface in three circles

Sculpture constructed by hand
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Volution Shells (Séquin 2003)

• Genus 0 and genus 1 generated by Surface Evolver

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Aesthetics of True Minimal Surfaces

• Large-area minimal surfaces are a challenge for
  any artist to improve on.
• For ribbon-like minimal surfaces, the artist
  typically prefers a deeper channel
• ? more drama and more strength.

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Part VI

• SNOWSCULPTING PLANS FOR 2004
• A realistic possibility a type of Volution
  shell.
• A really crazy ideaTurning a Snowball Inside
  Out ? ? ?
• ? Discussion of inadequacy of CAD tools

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Sphere Eversion

• In 1980, the blind mathematician B. Morin, (born
  1931) conceived of a way how a sphere can be
  turned inside-out
• Surface may pass through itself,
• but no ripping, puncturing, creasing
  allowed,e.g., this is not an acceptable solution

PINCH
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Morin Surface

• But there are more contorted paths that can
  achieve the desired goal.
• The Morin surface is the half-way point of one
  such path

John Sullivan The Optiverse
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Simplest Model

• Partial cardboard model based on the simplest
  polyhedral sphere ( cuboctahedron) eversion.

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Gridded Models for Transparency

• 3D-Print from Zcorp

SLIDE virtual model
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Shape Adaption for Snow Sculpture

• Restructured Morin surface to fit block size
  (10 x 10 x 12)

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Shape Optimization

• What is the fairest surface with the
  connectivity of the Morin surface that will fill
  the given bounding box ?
• Minimal surfaces are of no help, since this
  object clearly must have some positive curvature
  !
• What other functionals could we use ?Is there a
  Beauty Functional ?

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Beauty Functional Desirable Properties

• Smoothness continuous differentiability.
• Fairness even distribution of curvature
• Monotonicity preserving no unnecessary bulges,
  ripples.
• Invariance under rigid-body transforms, uniform
  scaling.
• Stability small change in specs ? small change
  in shape.
• Consistency no change if extra point is added on
  the shape.
• Technical relevance leads to spheres, cylinders,
  cones, tori.

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Various Optimization Functionals

• Minimum Length / Area (rubber bands, soap
  films)? Polygons -- Minimal Surfaces.
• Minimum Bending Energy (stiff Elastica) ? k2
  ds -- ? k12 k22 dA ? Splines
  -- Minimum Energy Surfaces.
• Minumum Curvature Variation (no natural model
  ?) ? (dk / ds)2 ds -- ? (dk1/ds)2 (dk2/ds)2
  dA ? Circles -- Cyclides Spheres, Cones,
  Tori ? Minumum Variation Surfaces (MVS)

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Minimum-Variation Surfaces
D4h
Genus 5
Oh
Genus 3

• The most pleasing smooth surfaces
• Constrained only by topology, symmetry, size.

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Optimization With Constraints

• Create the fairest possible surface that fits all
  the given constraints, which could be
• Position Give points to be interpolated
• Normals Define tangent planes
• Curvature Define a quadric to be matched
• Pictures based on implementation by Henry Moreton
  in 1993
• Used quintic Hermite splines for curves
• Used bi-quintic Bézier patches for surfaces
• Global optimization of all DoFs (slow!)

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Comparison MES ?? MVS(genus 4 surfaces)
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Comparison MES ?? MVS

• Things get worse for MES as we go to higher genus

Genus-5 MES
MVS
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Another ProblemMake Surface Transparent

• Realize surface as a grid.
• Draw a mesh of smooth lines onto the surface
• Ideally, these aregeodesic lines.

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Real Geodesics

• Chaotic Pathproduced by a geodesic lineon a
  surfacewith concaveas well as convex regions.

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Geodesic Lines

• Fairest curve is a straight line.
• On a surface, these are Geodesic lines (they
  bend with the given surface, but make no
  gratuitous lateral turns).
• We can easily draw such a curve from an initial
  point in a given direction
• Step-by-step construction of the next point (or
  of a short line segment).
• But connecting two given points on a given
  surface by a geodesic is an NP-hard problem.

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Another Use for Geodesics

• Map a complex graph onto a genus-3 surface
• Edges of graph should be nice, smooth curves.

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Strut Construction in Snow

• Drawing lines is not good enough for
  snow-sculpturewe need struts of substantial
  thickness.
• As few struts as possible should give a good
  viewof the whole smoothly curved surface.
• We will cut windows into a smooth surface, so
  that a network of struts is left standing.
• Surface of struts should follow curvature of
  surface,and their sides should be normal to the
  surface.
• How do we create a CAD model of this ?-- Some
  kind of sophisticated CSG operation ?
• Moreover, the struts in our model should
  beadjustable in width and in depth

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Best Modeling Effort as of 5/25/03
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Havent Found Suitable Tools yet

• We are struggling with subdivision surfaces and
  with sweeps along spline curves
• We have created our own tools in SLIDE (Scene
  Language for Interactive Dynamic Environments),
  a research system built in my group.
• SLIDE can create surface-grid representations,
  but only at the chosen sampling density. We need
  super-sampling to obtain curved struts.

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Conceptual Design (3D Sketching)

• E.g. creating a new form ( a Moebius bridge )
• CAD Tools are totally inadequate.
• Effective design ideation involves more than just
  the eyes and perhaps a (3D?) stylus.
• WANTEDfull-hand haptics (palm and fingers),
  whole body gestures,group interactions,

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The Holy Grail of a CAD System (for abstract,
geometric sculpture design)

• Combines the best of physical / virtual worlds
• No gravity ? no scaffolding needed
• Parts have infinite strength ? dont break
• Parts can be glued together and taken apart
• Beams may bend like perfect splines (or MVC)
• Surfaces may stretch like soap films (or MVS)
• Parts may emulate materials properties (sound).

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QUESTIONS ?DISCUSSION ?