Download - Graphics Lunch, Oct. 27, 2011
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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
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TopologyTopology
Shape does not matter -- only connectivity.
Surfaces can be deformed continuously.
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(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.
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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
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““Optiverse”Optiverse” Sphere Eversion Sphere Eversion
Turning a sphere inside-out in an “energy”-efficient way.
J. M. Sullivan, G. Francis, S. Levy (1998)
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Bad Torus EversionBad Torus Eversion
macbuse: Torus Eversionhttp://youtu.be/S4ddRPvwcZI
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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 !
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Legal Torus EversionLegal Torus Eversion
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End of Story ? … No !
These two tori cannot be morphed into one another!
Circular cross-section Figure-8 cross-section
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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:
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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)
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How to Construct a Torus, Step (1):How to Construct a Torus, Step (1):
Step (1): Roll a “tube”,join up red meridians.
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How to Construct a Torus, Step (2):How to Construct a Torus, Step (2): Step 2: Loop:
join up yellow parallels.
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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”
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Various Fancy ToriVarious Fancy Tori
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An Even Fancier TorusAn Even Fancier Torus
A bottle with an internal knotted passage
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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.
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4 Generic Representatives of Tori4 Generic Representatives of Tori
For the 4 different regular homotopy classes:
OO O8 8O 88
Characterized by: PROFILE / SWEEP
?
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Figure-8 Warp Introduces Twist!Figure-8 Warp Introduces Twist!
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(Cut) Tube, with Zero Torsion(Cut) Tube, with Zero Torsion
Note the end-to-end mismatch in the rainbow-colored stripes
Cut
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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
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Twisted ParameterizationTwisted Parameterization
How do we get rid of unwanted twist ?
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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?
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Torus Classification ?
Of which type are these tori ?
= ? = ?
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Un-warping a Circle with 720° TwistUn-warping a Circle with 720° Twist
Animation by Avik Das
Simulation of a torsion-resistant material
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Unraveling a Trefoil Knot
Animation by Avik Das
Simulation of a torsion-resistant material
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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
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Transformation MapTransformation Map
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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:
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A Handle / Tunnel Combination:A Handle / Tunnel Combination:
View along purple arrow
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Two Views of the Two Views of the ““Handle / TunnelHandle / Tunnel””
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““Handle / TunnelHandle / Tunnel”” on a Disk on a Disk Flip roles by closing surface
above or below the disk
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ParameterParameterSwapSwap
(Conceptual)(Conceptual)
illegal pinch-off points
fixed central
saddle point
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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
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Sphere EversionSphere Eversion
S. Levy, D. Maxwell, D. Munzner: Outside-In (1994)
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Dirac Belt TrickDirac Belt Trick
Unwinding a loop results in 360° of twist
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Outside-InOutside-In Sphere Eversion Sphere Eversion
S. Levy, D. Maxwell, D. Munzner: Outside-In (1994)
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A Legal Handle / Tunnel SwapA Legal Handle / Tunnel Swap
Let the handle-tunnel ride this process !
Undo unwanted eversion:
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Sphere Eversion Half-Way PointSphere Eversion Half-Way Point
Morin surface
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Torus with Knotted TunnelTorus with Knotted Tunnel
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Analyzing the Twist in the RibbonsAnalyzing the Twist in the Ribbons
The meridial circles are clearly not twisted.
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Analyzing the Twist in the RibbonsAnalyzing the Twist in the Ribbons
The knotted lines are harder to analyze Use a paper strip!
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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 !
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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
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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.
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World of Wild and Wonderful ToriWorld of Wild and Wonderful Tori
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Another Sculpture ?Another Sculpture ?
Torus with triangular profile, making two loops, with 360° twist
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Doubly-Looped ToriDoubly-Looped Tori
Step 1: Un-warping the double loop into a figure-8No change in twist !
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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
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Mystery Solved !Mystery Solved !
Dbl. loop, 360° twist Fig.8, 360° twist Untwisted circle
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Doubly-Rolled TorusDoubly-Rolled Torus
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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
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Mystery Solved !Mystery Solved !
Doubly-rolled torus w. 360° twist Untwisted type 88 torus
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Tori with CollarsTori with Collars
Torus may have more than one collar !
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Turning a Collar into 360° TwistTurning a Collar into 360° Twist
Use the move from “Outside-In” based on the Dirac Belt Trick,
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Legal Torus EversionLegal Torus Eversion
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Torus Eversion: Lower Half-SliceTorus Eversion: Lower Half-Slice
Arnaud Cheritat, Torus Eversion: Video on YouTube
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Torus Eversion SchematicTorus Eversion Schematic
Shown are two equatorials. Dashed lines have been everted.
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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!
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Parameter Swap Move ComparisonParameter Swap Move Comparison
New: We need to un-twist a lobe; movement through 3D space: adds E-twist !