Mathematics Summary
******An analytic representation of the Boy surface
********The different faces of the projective plane
****Click here
****Click here.
J.P.Petit and J.Souriau
: Note to the Comptes Rendus de l'Académie des Sciences de Paris, October 5, 1981, vol. 293 pp. 269-272. Starting from a construction of the Boy surface where the meridional curves are represented as a family of ellipses, a two-parameter representation is constructed:
), Y(
), Z(
(In French: pages 1 and 7)
J.P.Petit
: The projective plane is what you get when you glue a disk onto itself. This object cannot be embedded in R
. The Boy surface is an immersion of this object in R
. Other surfaces, including points "cuspidaux", such as the Cross-cap and the Roman surface of Steiner are other representations of the projective plane in R
, which are no longer immersions, since the cusp points are singular points. Starting from a transformation C "creation of cusp points" and its inverse C
"confluence of cusp points" we show how to go from the Cross Cap to the Boy surface, via the Roman surface of Steiner. Incidentally, this shows how to go from a "right" Boy to a "left" Boy. It also indicates how to permute the cusp points of a Cross-Cap.
(In French: pages 1, 13, 14, 15 and 16)
3 - Virtual Reality
: Have you ever dreamed of freely rotating a Steiner surface, a Moebius strip or a Boy surface between your fingers? If yes, first download Cosmoplayer, free software, then enjoy.
4 - Polyhedral version of the transformation of a Cross cap into a Boy surface, right or left (as you choose)
Polyhedral version of the central model of the sphere inversion.
Projects
J.P.Petit
: Sphere and torus inversions, full of animated gifs.
J.P.Petit
: The inversion of the cube (in preparation).
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