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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 !