Yvon-Villarceau Circle Equivalents on Dupin Cyclides

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Francesco De Comité: Yvon-Villarceau Circle Equivalents on Dupin Cyclides. In: Bridges 2015. Pages 253–258



A torus contains four families of circles: parallels, meridians and two sets of Yvon-Villarceau circles. Craftworks and artworks based on Yvon-Villarceau circles can be very attractive. Dupin cyclides are images of tori under sphere inversion, so they contain the images of the torus circles families. I applied operations that are known to create effective artworks on tori to Dupin cyclides, and proved them to be feasible. The regularity and the hidden complexity of the objects I obtained make them very attractive. Reviving the 19th century's tradition of mathematical models making, I printed several models, which can help in understanding their geometry. The tools I developed can be generalized to explore transformations of other mathematical objects under sphere inversion. This exploration is just at its beginning, but has already produced interesting new objects.

Extended Abstract


 author      = {Francesco De Comit\'e},
 title       = {Yvon-Villarceau Circle Equivalents on Dupin Cyclides},
 pages       = {253--258},
 booktitle   = {Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture},
 year        = {2015},
 editor      = {Kelly Delp, Craig S. Kaplan, Douglas McKenna and Reza Sarhangi},
 isbn        = {978-1-938664-15-1},
 issn        = {1099-6702},
 publisher   = {Tessellations Publishing},
 address     = {Phoenix, Arizona},
 note        = {Available online at \url{http://archive.bridgesmathart.org/2015/bridges2015-253.html }},
 url         = {http://de.evo-art.org/index.php?title=Yvon-Villarceau_Circle_Equivalents_on_Dupin_Cyclides&action },

Used References

[1] Torus/Villarceau Circles Slide-Together Pattern. https://www.flickr.com/photos/yoshinobu_ miyamoto/5555289192 (accessed 24/01/2015).

[2] V. Chandru, D. Dutta, and C.M. Hoffmann. On the Geometry of Dupin Cyclides. The Visual Computer, 5(5):277–290, 1989.

[3] Francesco De Comit´e. Circle Packing Explorations. In George W. Hart and Reza Sarhangi, editors, Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture, pages 399–402, Phoenix, Arizona, 2013. Tessellations Publishing. Available online at http://archive.bridgesmathart.org/2013/bridges2013-399.pdf (accessed 22/01/2015).

[4] Francesco De Comit´e. Cardioidal Variations. In George Hart Gary Greenfield and Reza Sarhangi, editors, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, pages 349–352, Phoenix, Arizona, 2014. Tessellations Publishing. Available online at http://archive.bridgesmathart.org/2014/bridges2014-349.html.

[5] Mar´ıa Garc´ıa Monera and Juan Monterde. Building a Torus with Villarceau Sections. Journal for Geometry and Graphics, 15(1):93–99, 2011.

[6] Lionel Garnier, Hichem Barki, Sebti Foufou, and Loic Puech. Computation of Yvon-Villarceau Circles on Dupin Cyclides and Construction of Circular Edge Right Triangles on Tori and Dupin Cyclides. Computers & Mathematics with Applications, 68(12):1689–1709, 2014.

[7] George W. Hart. Slide-Together Geometric Paper Constructions. http://www.georgehart.com/ slide-togethers/slide-togethers.html (accessed 07/04/2015).

[8] Gabriela Ligenza. Gabriela Ligenza’s website. http:www.gabrielaligenza.com (accessed 27/01/2015).

[9] Michael Schrott and Boris Odehnal. Ortho-Circles of Dupin Cyclides. Journal for Geometry and Graphics, 10(1):73,98, 2006.


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