Schloss Dagstuhl, September 2014Schloss Dagstuhl, September 2014

Shape Representation

Carlo H. Séquin

University of California, Berkeley

“LEGO Knot” and an Optimization Problem in a High-Dimensional Discrete Solution Space

Discussion Points:Discussion Points:

Shape representation issuesat the start and conclusion

of designing RP models

Focus on HCI difficulties and CAD problems,at the start and end of a design / modeling project:

How to get started? How to get your ideas into the CAD system.

How to finish? How to get your model properly 3D printed.

User-Guided Inverse 3D Modeling

Few designs start from scratch in a vacuum. Often there is a previous artifact that provides inspiration or may even be close enough so that some high-level redesign might be the most effective approach. Unfortunately there may be no CAD files available or they may be at such a low level (100’000 triangles) that it is not a good starting point for a major redesign.

“User-Guided Inverse 3D Modeling” is an approach to re-create a well-structured, high-level, parameterized, procedural description of some geometry very close to the inspirational artifact. Its hierarchical structure and the degree of its parameterization are imposed with some high-level instructions by the designer, so that the resulting description is most appropriate to make the intended design changes.

REF: http://www.cs.berkeley.edu/~sequin/UGI3DM/index.html

Another Issue . . . Another Issue . . .

“LEGO®” Knots

EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley

Carlo H. Séquin

Inspiration: Henk van PuttenInspiration: Henk van Putten

“Borsalino” “Interaction”

Sculptural forms put together from a few modular shapes

Geometry of the Geometry of the BorsalinoBorsalino

Just 2 geometrical components: 3 semi-circular end-caps (orange) 6 curved connectors, bending through 45º== a square cross section swept along 9 circular arcs.

The Wonders of Rapid-PrototypingThe Wonders of Rapid-Prototyping

Two modular components can form the Borsalino

Connector R=2.4142

End-Cap R=1.0

Hands-on SculptingHands-on Sculpting

André Eveline Lorenzo

Inspiration: Jon KrawczykInspiration: Jon Krawczyk

303 2nd Street, San Francisco

Inspiration: Paul BlochInspiration: Paul Bloch

“After Wright” (Guggenheim, NYC)

LEGOLEGO®® DUPLO DUPLO

Match interface

More More ““User-StudiesUser-Studies”” (3 (3rdrd Gen.) Gen.)

Sienna (5) and Elise (7)

My Personal Quest:My Personal Quest:

What kind of parts does it takeWhat kind of parts does it taketo make to make Mathematical KnotsMathematical Knotswith nice graceful curvaturewith nice graceful curvatureand smooth loop closure ?and smooth loop closure ?

Real Knots: Trefoil (3_1)Real Knots: Trefoil (3_1)

One custom-designed piece (magenta) for smooth closure

D3 symmetry

Trefoil Knot (3_1)Trefoil Knot (3_1)

Real Knots: Figure-8 Knot (4_1)Real Knots: Figure-8 Knot (4_1)

Two new pieces (magenta, red) for smooth closure

4-fold glide symmetry

Figure-8 Knot (4_1)Figure-8 Knot (4_1)

Composition Problems Composition Problems in a Discrete Solution Spacein a Discrete Solution Space

Similar to the Zome-Tool Approximation:

Suppose we restrict ourselves to just using one single module!

Can we build elegant and symmetrical knots?

Single-Module KnotsSingle-Module Knots

Richard Zawitz:Richard Zawitz: Museum Tangle (1982) Museum Tangle (1982)

Pliable UnKnot made from 18 quarter-torus segments

Naef Wooden ToysNaef Wooden Toys

CaterpillarCaterpillar

Knots Made from ONE ModuleKnots Made from ONE Module

M. Zawidzki & K. Nishinari:

Problems: too many elements, lack of symmetry, self-intersections, bad loop closure.

Forming Closed Loops Is Difficult !Forming Closed Loops Is Difficult !M. Zawidzki & K. Nishinari:

A First Try on a Figure-8 KnotA First Try on a Figure-8 Knot

Composed of 4×10 wedge elements (4-fold symmetry)

Does not properly close! ( 6 DoF: x, y, z, 3 angles )

12-gon profile

The Module ChosenThe Module Chosen

16-gon cross section (finer control of azimuth)

30°bending angle in module (fewer overall modules)

r/R = 0.3 (pipe radius / bending radius “wiggle space”)

A First Batch of 20 ModulesA First Batch of 20 Modules

Out of the “Uprint” FDM machine

Simplest Simplest ““ModKnotModKnot””: K 3_1: K 3_1

Trefoil Knot: K 3_1; D3 symmetry; uses 33 modules

Trefoil Knot SculptureTrefoil Knot Sculpture

33 parts

Simple Simple ““ModKnotsModKnots””: K 4_1: K 4_1

Figure-8 Knot: K 4_1; S4 symmetry; uses 40 modules

Even with the computer plan, this is difficult to assemble!

Figure-8 Knot SculptureFigure-8 Knot Sculpture

40 parts

Cinquefoil Knot: K 5_1Cinquefoil Knot: K 5_1

D5 symmetry

50 modules

Mathematical LinksMathematical Links

Borromean Link

Forming Mathematical Knots and LinksForming Mathematical Knots and Links

Challenging Computational Issue:

What is the right algorithm to find thebest solution for any such problem in its high-dimensional solution space ?