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 by gluing 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" it is shown how one can 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 is also indicated 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 so, 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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