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Exploring Torus Homotopies: A Study in Topology and Shape Deformation

Discover the world of Torus homotopies in this engaging study encompassing smooth deformations, legal eversion, surfaces parameterization, and twist manipulations. Uncover the beauty and complexity of Tori shapes!

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Exploring Torus Homotopies: A Study in Topology and Shape Deformation

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  1. Graphics Lunch, Oct. 27, 2011 “Tori Story”(Torus Homotopies ) Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. Topology • Shape does not matter -- only connectivity. • Surfaces can be deformed continuously.

  3. (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.

  4. Smoothly 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

  5. “Optiverse” Sphere Eversion J. M. Sullivan, G. Francis, S. Levy (1998) Turning a sphere inside-out in an “energy”-efficient way.

  6. Bad Torus Eversion macbuse: Torus Eversion http://youtu.be/S4ddRPvwcZI

  7. Illegal Torus Eversion • Moving the torus through a puncture is not legal. ( If this were legal, then everting a sphere would be trivial! ) NO !

  8. Legal Torus Eversion

  9. End of Story ? … No ! Circular cross-section Figure-8 cross-section • These two tori cannot be morphed into one another!

  10. Tori Can Be Parameterized • Surface decorations (grid lines) are relevant. • We want to maintain them during all transformations. Orthogonalgrid lines: These 3 tori cannot be morphed into one another!

  11. 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)

  12. How to Construct a Torus, Step (1): • Step (1): Roll a “tube”,join up red meridians.

  13. How to Construct a Torus, Step (2): • Step 2: Loop:join up yellowparallels.

  14. Surface 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”

  15. Various Fancy Tori

  16. A bottle with an internal knotted passage An Even Fancier Torus

  17. Tori 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.

  18. 4 Generic Representatives of Tori • For the 4 different regular homotopy classes: ? OO O8 8O 88 Characterized by: PROFILE / SWEEP

  19. Figure-8 Warp Introduces Twist!

  20. (Cut) Tube, with Zero Torsion Cut Note the end-to-end mismatch in the rainbow-colored stripes

  21. 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

  22. Twisted Parameterization How do we get rid of unwanted twist ?

  23. Dealing with a Twist of 360° Take a regular torus of type “OO”, and introduce meridial twist of 360°, What torus type do we get? “OO” + 360°M-twist warp thru 3D  representative “O8”

  24. Torus Classification ? = ? = ? Of which type are these tori ?

  25. Un-warping a Circle with 720° Twist Simulation of a torsion-resistant material Animation by Avik Das

  26. Unraveling a Trefoil Knot Simulation of a torsion-resistant material Animation by Avik Das

  27. 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

  28. Transformation Map

  29. Trying to Swap Parameters This is the goal: Focus on the area where the tori touch, and try to find a move that flips the surface from one torus to the other.

  30. A Handle / Tunnel Combination: View along purple arrow

  31. Two Views of the “Handle / Tunnel”

  32. Flip roles by closing surface above or below the disk “Handle / Tunnel” on a Disk

  33. ParameterSwap(Conceptual) fixed central saddle point illegal pinch-off points

  34. Flipping the Closing Membrane • Use a classical sphere-eversion process to get the membrane from top to bottom position! Starting Sphere Everted Sphere

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

  36. Dirac Belt Trick Unwinding a loop results in 360° of twist

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

  38. Undo unwanted eversion: A Legal Handle / Tunnel Swap Let the handle-tunnel ride this process !

  39. Sphere Eversion Half-Way Point Morin surface

  40. Torus with Knotted Tunnel

  41. Analyzing the Twist in the Ribbons The meridial circles are clearly not twisted.

  42. Analyzing the Twist in the Ribbons The knotted lines are harder to analyze  Use a paper strip!

  43. Torus Eversion Half-Way Point This would make a nice constructivist sculpture ! What is the most direct move back to an ordinary torus ?

  44. Conclusions • 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 toripromising intriguing constructivist sculptures. • Ways to analyze and classify such weird tori.

  45. 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.

  46. World of Wild and Wonderful Tori

  47. Another Sculpture ? Torus with triangular profile, making two loops, with 360° twist

  48. Doubly-Looped Tori Step 1: Un-warping the double loop into a figure-8No change in twist !

  49. Movie: Un-warping a Double Loop Simulation of a material with strong twist penalty “Dbl. Loop with 360° Twist” by Avik Das

  50. Mystery Solved ! Dbl. loop, 360° twist  Fig.8, 360° twist  Untwisted circle

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