Graphics Lunch, Oct. 27, 2011Graphics Lunch, Oct. 27, 2011

“Tori Story” ( Torus Homotopies )

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

Carlo H. Séquin

TopologyTopology

Shape does not matter -- only connectivity.

Surfaces can be deformed continuously.

(Regular) Homotopy(Regular) Homotopy

Two shapes are called homotopic, if they can be transformed into one anotherwith a continuous smooth deformation(with no kinks or singularities).

Such shapes are then said to be:in the same homotopy class.

Smoothly Deforming SurfacesSmoothly Deforming Surfaces

Surface may pass through itself.

It cannot be cut or torn; it cannot change connectivity.

It must never form any sharp creases or points of infinitely sharp curvature.

OK

““Optiverse”Optiverse” Sphere Eversion Sphere Eversion

Turning a sphere inside-out in an “energy”-efficient way.

J. M. Sullivan, G. Francis, S. Levy (1998)

Bad Torus EversionBad Torus Eversion

macbuse: Torus Eversionhttp://youtu.be/S4ddRPvwcZI

Illegal Torus EversionIllegal Torus Eversion

Moving the torus through a puncture is not legal.

( If this were legal, then everting a sphere would be trivial! )

NO !

Legal Torus EversionLegal Torus Eversion

End of Story ? … No !

These two tori cannot be morphed into one another!

Circular cross-section Figure-8 cross-section

Tori Can Be Parameterized

These 3 tori cannot be morphed into one another!

Surface decorations (grid lines) are relevant.

We want to maintain them during all transformations.

Orthogonalgrid lines:

What is a Torus?What is a Torus?

Step (1): roll rectangle into a tube.

Step (2): bend tube into a loop.

magenta “meridians”, yellow “parallels”, green “diagonals” must all close onto themselves!

(1) (2)

How to Construct a Torus, Step (1):How to Construct a Torus, Step (1):

Step (1): Roll a “tube”,join up red meridians.

How to Construct a Torus, Step (2):How to Construct a Torus, Step (2): Step 2: Loop:

join up yellow parallels.

Surface Decoration, ParameterizationSurface Decoration, Parameterization Parameter grid lines must close onto themselves.

Thus when closing the toroidal loop, twist may be added only in increments of ±360°

+360° 0° –720° –1080°Meridial twist , or “M-twist”

Various Fancy ToriVarious Fancy Tori

An Even Fancier TorusAn Even Fancier Torus

A bottle with an internal knotted passage

Tori Story: Main MessageTori Story: Main Message Regardless of any contorted way

in which one might form a decorated torus, all possible tori fall into exactly four regular homotopy classes.[ J. Hass & J. Hughes, Topology Vol.24, No.1, (1985) ]Oriented surfaces of genus g fall into 4g homotopy classes.

All tori in the same class can be deformed into each other with smooth homotopy-preserving motions.

I have not seen a side-by-side depiction of 4 generic representatives of the 4 classes.

4 Generic Representatives of Tori4 Generic Representatives of Tori

For the 4 different regular homotopy classes:

OO O8 8O 88

Characterized by: PROFILE / SWEEP

?

Figure-8 Warp Introduces Twist!Figure-8 Warp Introduces Twist!

(Cut) Tube, with Zero Torsion(Cut) Tube, with Zero Torsion

Note the end-to-end mismatch in the rainbow-colored stripes

Cut

Twist Is Counted Modulo 720°Twist Is Counted Modulo 720° We can add or remove twist in a ±720° increment

with a “Figure-8 Cross-over Move”.

Push the yellow / green ribbon-crossing down through the Figure-8 cross-over point

Twisted ParameterizationTwisted Parameterization

How do we get rid of unwanted twist ?

Dealing with a Twist of 360Dealing with a Twist of 360°

“OO” + 360°M-twist warp thru 3D representative “O8”

Take a regular torus of type “OO”,and introduce meridial twist of 360°,What torus type do we get?

Torus Classification ?

Of which type are these tori ?

= ? = ?

Un-warping a Circle with 720° TwistUn-warping a Circle with 720° Twist

Animation by Avik Das

Simulation of a torsion-resistant material

Unraveling a Trefoil Knot

Animation by Avik Das

Simulation of a torsion-resistant material

Other Tori Transformations ?

Eversions:

Does the Cheritat operation work for all four types?

Twisting:

Twist may be applied in the meridial direction or in the equatorial direction.

Forcefully adding 360 twist may change the torus type.

Parameter Swap:

Switching roles of meridians and parallels

Transformation MapTransformation Map

Trying to Swap ParametersTrying to Swap Parameters

Focus on the area where the tori touch, and try to find a move that flips the surface from one torus to the other.

This is the

goal:

A Handle / Tunnel Combination:A Handle / Tunnel Combination:

View along purple arrow

Two Views of the Two Views of the ““Handle / TunnelHandle / Tunnel””

““Handle / TunnelHandle / Tunnel”” on a Disk on a Disk Flip roles by closing surface

above or below the disk

ParameterParameterSwapSwap

(Conceptual)(Conceptual)

illegal pinch-off points

fixed central

saddle point

Flipping the Closing MembraneFlipping the Closing Membrane

Use a classical sphere-eversion process to get the membrane from top to bottom position!

Everted Sphere

Starting Sphere

Sphere EversionSphere Eversion

S. Levy, D. Maxwell, D. Munzner: Outside-In (1994)

Dirac Belt TrickDirac Belt Trick

Unwinding a loop results in 360° of twist

Outside-InOutside-In Sphere Eversion Sphere Eversion

S. Levy, D. Maxwell, D. Munzner: Outside-In (1994)

A Legal Handle / Tunnel SwapA Legal Handle / Tunnel Swap

Let the handle-tunnel ride this process !

Undo unwanted eversion:

Sphere Eversion Half-Way PointSphere Eversion Half-Way Point

Morin surface

Torus with Knotted TunnelTorus with Knotted Tunnel

Analyzing the Twist in the RibbonsAnalyzing the Twist in the Ribbons

The meridial circles are clearly not twisted.

Analyzing the Twist in the RibbonsAnalyzing the Twist in the Ribbons

The knotted lines are harder to analyze Use a paper strip!

Torus Eversion Half-Way PointTorus Eversion Half-Way Point

What is the most direct move back to an ordinary torus ?This would make a nice constructivist sculpture !

Just 4 Tori-Classes!

Four Representatives:

Any possible torus fits into one of those four classes!

An arsenal of possible moves.

Open challenges: to find the most efficent / most elegant trafo (for eversion and parameter swap).

A glimpse of some wild and wonderful tori promising intriguing constructivist sculptures.

Ways to analyze and classify such weird tori.

ConclusionsConclusions

Q U E S T I O N S ?Q U E S T I O N S ?

Thanks:

John Sullivan, Craig Kaplan, Matthias Goerner;Avik Das.

Our sponsor: NSF #CMMI-1029662 (EDI)

More Info:

UCB: Tech Report EECS-2011-83.html

Next Year:

Klein bottles.

World of Wild and Wonderful ToriWorld of Wild and Wonderful Tori

Another Sculpture ?Another Sculpture ?

Torus with triangular profile, making two loops, with 360° twist

Doubly-Looped ToriDoubly-Looped Tori

Step 1: Un-warping the double loop into a figure-8No change in twist !

Movie: Un-warping a Double Loop Movie: Un-warping a Double Loop Simulation of a material with strong twist penalty

“Dbl. Loop with 360° Twist” by Avik Das

Mystery Solved !Mystery Solved !

Dbl. loop, 360° twist Fig.8, 360° twist Untwisted circle

Doubly-Rolled TorusDoubly-Rolled Torus

Double Roll Double Roll Double Loop Double Loop Reuse a previous figure, but now with double walls:

Switching parameterization: Double roll turns into a double loop; The 180° lobe-flip removes the 360° twist; Profile changes to figure-8 shape; Unfold double loop into figure-8 path. Type 88

Mystery Solved !Mystery Solved !

Doubly-rolled torus w. 360° twist Untwisted type 88 torus

Tori with CollarsTori with Collars

Torus may have more than one collar !

Turning a Collar into 360° TwistTurning a Collar into 360° Twist

Use the move from “Outside-In” based on the Dirac Belt Trick,

Legal Torus EversionLegal Torus Eversion

Torus Eversion: Lower Half-SliceTorus Eversion: Lower Half-Slice

Arnaud Cheritat, Torus Eversion: Video on YouTube

Torus Eversion SchematicTorus Eversion Schematic

Shown are two equatorials. Dashed lines have been everted.

A Different Kind of MoveA Different Kind of Move

Start with a triple-fold on a self-intersecting figure-8 torus;

Undo the figure-8 by moving branches through each other;

The result is somewhat unexpected:

Circular Path, Fig.-8 Profile, Swapped Parameterization!

Parameter Swap Move ComparisonParameter Swap Move Comparison

New: We need to un-twist a lobe; movement through 3D space: adds E-twist !