Solid Modeling Symposium, Seattle 2003

1
Solid Modeling Symposium, Seattle 2003

• Aesthetic EngineeringCarlo H. Séquin
• EECS Computer Science Division
• University of California, Berkeley

2
I Am Not an Artist
3
I am a Designer, Engineer
CCD Camera, Bell Labs, 1973 Soda Hall,
Berkeley, 1994
RISC chip, Berkeley, 1981 Octa-Gear,
Berkeley, 2000
4
Artistic Geometry

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

5
What Drives My Research ?

• Whatever I need most urgently to get a real job
  done.
• Most of my jobs involve building things-- not
  just pretty pictures on a CRT.
• Today Report on some ongoing activities --
  motivation and progress so far.
• Thanks to Ling Xiao, Ryo Takahashi,Alex
  Kozlowski.

6
Outline Three Defining Tasks

• 1 Mapping graphs onto surfaces of suitable
  genus with a high degree of symmetry.
• 2 Making models of self-intersecting surfaces
  such as Klein-bottles, Boy Surface, Morin
  Surface
• 3 Coming up with an interesting and doable
  design for a snow-sculpture for January 2004.

7
Outline Some Common Problems

• A Which is the fairest (surface) of them all
  ?
• B Drawing geodesic lines (or curves with
  linearly varying curvature) between two
  points on a surface.
• C Making gridded surface representations
  (different needs for different applications).

8
TASK GROUP 1Two Graph-Mapping
Problems(courtesy of Prof. Jürgen Bokowski)

• Given some abstract graph
• K12 complete graph with 12 vertices,
• Dyck Graph (12vertices, but only 48 edges)
• Embed each of these graphs crossing-free
• in a surface with lowest possible genus,
• so that an orientable matroid results,
• maintaining as much symmetry as possible.

9
Graph K12
10
Mapping Graph K12 onto a Surface(i.e., an
orientable two-manifold)

• Draw complete graph with 12 nodes
• Has 66 edges
• Orientable matroid has 44 triangular facets
• Euler E V F 2 2Genus
• 66 12 44 2 12 ? Genus 6
• ? Now make a (nice) model of that !

11
Bokowskis Goose-Neck Model
12
Bokowskis ( Partial ) Virtual Model on a
Genus 6 Surface
13
My Model

• Find highest-symmetry genus-6 surface,
• with convenient handles to route edges.

14
My Model (cont.)

• Find suitable locations for twelve vertices
• Maintain symmetry!
• Put nodes at saddle points, because of 11
  outgoing edges, and 11 triangles between them.

15
My Model (3)

• Now need to place 66 edges
• Use trial and error.
• Need a 3D model !
• No nice CAD model yet.

16
A 2nd Problem Dycks Graph

• 12 vertices,
• but only 48 edges.
• E V F 2 2Genus
• 48 12 32 2 6 ? Genus 3

17
Another View of Dycks Graph

• Difficult to connect up matching nodes !

18
Folding It into a Self-intersecting Polyhedron
19
Towards a 3D Model

• Find highest-symmetry genus-3 surface? Klein
  Surface (tetrahedral frame).

20
Find Locations for Vertices

• Actually harder than in previous example, not
  all vertices connected to one another. (Every
  vertex has 3 that it is not connected to.)
• Place them so that themissing edges do not
  break the symmetry
• ? Inside and outside on each tetra-arm.
• Do not connect the vertices that lie on thesame
  symmetry axis(same color)(or this one).

21
A First Physical Model

• Edges of graph should be nice, smooth curves.

Quickest way to get a model Painting a
physical object.
22
What Are the CAD Tasks Here ?

• 1) Make a fair surface of given genus.
• 2) Symmetrically place vertices on it.
• 3) Draw geodesic lines between points.
• 4) Color all regions based on symmetry.
• ? Lets address tasks 1) and 3)

23
Construction of Fair Surfaces

• Input Genus, symmetry class, size
• Output Fairest surface possible
• Highest symmetry G3 ? Tetrahedral
• Smooth Gn continuous (n?2)
• Simple No unnecessary undulations
• Good parametrization (for texturing)
• Representation Efficient, for visualization, RP
• ? Use some optimization process
• ? Is there a Beauty Functional ?

24
Various Optimization Functionals

• Minimum Length / Area (rubber bands, soap
  films)? Polygons -- Minimal Surfaces.
• Minimum Bending Energy (thin plates,
  Elastica) ? k2 ds -- ? k12 k22
  dA ? Splines -- Minimum Energy Surfaces.
• Minumum Curvature Variation (no natural model
  ?) ? (dk / ds)2 ds -- ? (dk1/de1)2 (dk2/de2)2
  dA ? Circles -- Cyclides Spheres,
  Cones, Tori ? Minumum Variation Curves /
  Surfaces (MVC, MVS)

25
Minimum-Variation Surfaces
D4h
Oh
Genus 3
Genus 5

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

26
Comparison MES ?? MVS(genus 4 surfaces)
27
Comparison MES ?? MVS

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

pinch off
3 holes
Genus-5 MES
MVS
28
1st Implementation Henry Moreton

• Thesis work by Henry Moreton in 1993
• Used quintic Hermite splines for curves
• Used bi-quintic Bézier patches for surfaces
• Global optimization of all DoFs (many!)
• Triply nested optimization loop
• Penalty functions forcing G1 and G2 continuity
• ? SLOW ! (hours, days!)
• But results look very good

29
What Can Be Improved?

• Continuity by construction
• E.g., Subdivision surfaces
• Avoids need for penalty functions
• Improves convergence speed (gt100x)
• Hierarchical approach
• Find rough shape first, then refine
• Further improves speed (gt10x)
• Computers are 100x faster than 1993
• ? gt105 ? Days become seconds !

30
B Drawing onto that Surface

• MVS gives us a good shape for the surface.
• Now we want to draw nice, smooth curvesThey
  look like geodesics

31
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 (one
  line segment per polyhedron facet).

32
Real Geodesics

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

33
Geodesic Line Between 2 Points
T
S

• Connecting two given points with the shortest
  geodesic on a high-genus surface is an NP-hard
  problem.

34
Try Target-Shooting

• Send geodesic path from S towards T
• Vary starting direction do binary search for
  hit.

35
Target-Shooting Problem (2)

• Where Gauss curvature lt 0 (saddle regions)
• ? no (stable) path ? defocussing effect.

T1
T2
S
T2
V
S
V
T
T1
T1, T2 can only be reached by going through V !
36
Polyhedral Angle Ambiguity

• At non-planar vertices in a polyhedral surface
  there is an angle deficit (Ggt0) or excess (Glt0).
• Whenever a path hits a vertex,we can choose
  within this angle,how the path should continue.
• If, in our binary search for a target hit,the
  path steps across a vertex,we can lock the path
  to that vertex,and start a new shooting game
  from there.

37
Pseudo Geodesics

• Need more control than geodesics can offer.
• Want to space the departing curves from a vertex
  more evenly, avoid very acute angles.
• Need control over starting and ending tangent
  directions (like Hermite spline).

38
LVC Curves (instead of MVC)

• Curves with linearly varying curvaturehave two
  degrees of freedom kA kB,
• Allows to set two additional parameters,i.e.,
  the start / ending tangent directions.

CURVATURE
kB
ARC-LENGTH
kA
B
A
39
The Complete Shooting Game

• Alternate shooting from both ends,
• gradually adjusting the two end-curvature
  parameters until the two points are connected and
  the two specified tangent directions are met.
• Need to worry about angle ambiguity,whenever the
  path correction jumpsover a vertex of the
  polyhedron.
• Gets too complicated instabilities
• gt NOT RECOMMENDED !

40
More Promising Approach to Findinga Geodesic
LVC Connection

• Assume, you already have some path that connects
  the two points with the desired route on the
  surface (going around the right handles).
• Move all the facet edge crossing points so as to
  even out the curvature differences between
  neighboring path sample pointswhile approaching
  the LVC curve with the desired start / end
  tangents.

41
Path-Optimization towards LVC

• Locally move locations of edge crossingsso as to
  even out variation of curvature

S
C
V
C
T
As path moves across a vertex, re-analyze the
gradient on the new edges, and exploit angle
ambiguity.
42
TASK GROUP 2Making RP Models of Math Surfaces

• Klein Bottles
• Boys Surface
• Morin Surface
• Intriguing, self-intersecting in 3D

43
Skeleton of Klein Bottle

• Transparency in the dark old ages when I could
  only make BW prints
• Take a grid-approach to depicting transparent
  surfaces.
• Need to find a good parametrization,which
  defines nicely placed grid lines.
• Ideally, avoid intersections of struts (not
  achieved in this figure).

SEQUIN, 1981
44
Triply Twisted Figure-8 Klein Bottle
SEQUIN 2000

• Strut intersections can be avoided by design
  because of simplicity of intersection line and
  regularity of strut crossings.

45
Avoiding Self-intersections

• Rectangular surface domain of Klein bottle.
• Arrange strut patternas shown on the left.
• After the figure-8 fold, struts pass smoothly
  through one another.

46
A Look into the FDM Machine
47
Triply Twisted Figure-8 Klein Bottle
As it comes out of the FDM machine
48
The Finished Klein Bottle (supports removed)
49
The Projective Plane
PROJECTIVE PLANE
C
-- Walk off to infinity -- and beyond come
back upside-down from opposite direction. Project
ive Plane is single-sided has no edges.
50
Model of Boy Surface

• Computer graphics by John Sullivan (1998)

51
Double Covering of Boy Surface

• Wire model byCharles Pugh( 1980 )
• Decorated by C. H. Séquin
• Equator
• 3 Meridians, 120º apart

52
Can We Avoid Strut Intersectionsfor Boys
Surface ?

• This is much harder
• More difficult to find a nice, regularly gridded
  parametrization,
• Intersection lines are more complicated,
• Harder to predict where parameter lines will
  cross over.

53
Tessellation from Surface Evolver

• Triangulation from optimal polyhedron.
• Mesh dualization.
• Strut thickening.
• FDM fabrication.
• Quad facet !
• Intersecting struts.

54
Paper Model with Regular Tiles

• Only vertices of valence 3.
• Only meshes with 5, 6, or 7 sides.
• Struts pass through holes.--gt Permits the use
  of a modular component...

55
A Modular Triconnector

• Prototype made in the FDM machine

56
Assembly of the Tiled Boy Surface
KIHA LEE
57
(No Transcript)
58
Boy Surface in Oberwolfach

• Sculpture constructed by Mercedes Benz
• Photo courtesy John Sullivan

59
TASK GROUP 3Combining Math Model Makingwith
some artistic ambitions

• This needs some background

60
Brent Collins
Hyperbolic Hexagon II
61
Brent Collins Stacked Saddles
62
Scherks 2nd Minimal Surface
Normal biped saddles
Generalization to higher-order saddles(monkey
saddle)
63
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 ?

64
Closing the Loop
straight or twisted
65
Brent Collins Prototyping Process
Mockup for the "Saddle Trefoil"
Armature for the "Hyperbolic Heptagon"
Time-consuming ! (1-3 weeks)
66
Sculpture Generator I, GUI
67
V-art
VirtualGlassScherkTowerwith MonkeySaddles(R
adiance 40 hours) Jane Yen
68
Collins Fabrication Process
Wood master patternfor sculpture
Layered laminated main shape
Example Vox Solis
69
Slices through Minimal Trefoil
50
10
23
30
45
5
20
27
35
2
15
25
70
Profiled Slice through Heptoroid

• One thick slicethru sculpture,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.

71
Emergence of the Heptoroid (1)
Assembly of the precut boards
72
Emergence of the Heptoroid (2)
Forming a continuous smooth edge
73
Emergence of the Heptoroid (3)
Smoothing the whole surface
74
The Finished Heptoroid

• at Fermi Lab Art Gallery (1998).

75
Various Scherk-Collins Sculptures
76
Hyper-Sculpture Family of 12 Trefoils
W2
W1
B1 B2 B3
B4
77
Cohesion

• SIGGRAPH2004 Art Gallery

78
Stan Wagon, Macalester College, St. Paul, MN

• Leader of Team USA Minnesota

79
Snow-Sculpting, 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)

80
Breckenridge, CO, 1999

• Helaman Ferguson Invisible Handshake

81
Breckenridge, 2000

• Robert Longhurst
• Rhapsody in White
• 2nd Place

82
Monkey Saddle Trefoil

• from Sculpture Generator I

83
Annual Championships in Breckenridge, CO
84
(No Transcript)
85
Day 1 The Monolith

• Cut away prisms

86
End of Day 2

• The Torus

87
Day 3, 4 Carving the Flanges, Holes
88
Day 5, am Surface Refinement
89
Whirled White Web
90
(No Transcript)
91
1240 pm -- 42 F
92
1241 pm -- 42 F
93
124001
Photo StRomain
94
3 pm WWW Wins Silver Medal
95
Snow-Sculpting Plans for 2004

• Turning a Snowball Inside Out
• Design is due July 1, 2003
• Again, I am having some problemsmaking a good
  CAD model.

96
Sphere Eversion

• 1960, 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
97
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
98
Simplest Model

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

99
Gridded Models for Transparency

• 3D-Print from Zcorp

SLIDE virtual model
100
Shape Adaption for Snow Sculpture

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

101
Make Surface Transparent

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

102
Best Modeling Effort as of 5/25/03

• Used Sweep-Morph for best controlof placing
  parameter lines.
• Developed a special offset-surface generator that
  cuts windows into all the facets,so that only
  a grid structure remains.

103
Latest FDM Model 6/1/03

• Work to Be Done
• Need a perfect CAD model for bronze cast.
• Struts should be curved and follow surface.
• Should be of uniform thickness.
• Could involve challenging CSG operation.
• Plan Build into offset-surface generator.

104
CAD and Modeling Tools

• State of the art is lacking
• Fairly generic utilities are missing
• Surface optimization,
• Geodesic lines,
• Gridded surface representations.
• We are building our own procedural extensions to
  fill this void.

105
Tools for Early Conceptual Design

• For creating new forms, e.g. a Moebius bridge
• 3D Sketching Tools are totally inadequate.
• I typically find myself using cardboard, wires,
  scotch-tape, styrofoam, clay, wiremesh
• Effective design ideation involves more than just
  the eyes and perhaps a (3D?) stylus.

106
My Dream of a CAD System (for abstract,
geometric sculpture design)

• Combines the best of virtual / physical worlds
• No gravity ? no scaffolding needed,
• Parts have infinite strength ? dont break,
• Parts can be glued together and taken apart.
• Has built-in optimization functionality
• Beams may bend like steel wires (or MVC),
• Surfaces may stretch like soap films (or MVS),
• Geodesic threads on surfaces.
• Provides a hands-on feel during modeling
  process.
• As much co-located haptic feedback as possible.

107
Conclusions

• A glimpse of research in progress,
• what motivates me and my students,
• and how we tackle some practical problems.
• This is a solicitation for help with
• references to similar work,
• suggestions of better approaches,
• or outright collaboration.

108
QUESTIONS ?DISCUSSION ?