Consolidated document. 2 original pages have been brought together here; the links below open each section.
Find: Scientific documents.
Contents
1-J.P.Petit and J.Souriau: An analytical representation of Boy's surface: Note to the Comptes Rendus de l'Académie des Sciences de Paris, October 5, 1981, vol. 293 pp. 269-272. From a construction of Boy's surface in which the meridian curves are represented as a family of ellipses, a two-parameter representation is constructed:
X(m,q), Y(m,q), Z(m,q)
(In French: - pages 1 and 7)
2-J.P.Petit: The different faces of the projective plane: The projective plane is what you get by gluing a disk onto itself. This object cannot be embedded in R3. Boy's surface is an immersion of this object in R3. Other surfaces, comprising "cuspidal" points, such as the Cross-cap and Steiner's Roman surface, are other representations of the projective plane in R3, which are no longer immersions, since the cuspidal points are singular points. From a transformation C "creation of cuspidal points" and its inverse C-1 "confluence of cuspidal points" it is shown how one can go from the Cross Cap to Boy's surface, via Steiner's Roman surface. Incidentally this shows how to go from a "right" Boy to a "left" Boy. It is also indicated how to permute the cuspidal points of a Cross-Cap. (In French: pages 1, 13, 14, 15 and 16)
3 - Virtual Reality: Have you ever dreamed of twirling a Steiner surface, a Moebius strip or a Boy's surface between your fingers at will? If so, first download Cosmoplayer, as freeware, then enjoy.
4 - polyhedral version of the transformation of a Cross cap into a Boy's Surface, right or left (your choice)Click here
5 -Polyhedral version of the central model of the sphere eversion.Click here.
Projects:
J.P.Petit: The eversions of the sphere and the torus, full of animated gifs.
J.P.Petit: The eversion of the cube (in preparation).
Continuation
1-J.P.Petit and J.Souriau: An analytical representation of Boy's surface: Note to the Comptes Rendus de l'Académie des Sciences de Paris, October 5, 1981, vol. 293 pp. 269-272. From a construction of Boy's surface in which the meridian curves are represented as a family of ellipses, a two-parameter representation is constructed:
X(m,q), Y(m,q), Z(m,q)
(In French: - pages 1 and 7)
2-J.P.Petit: The different faces of the projective plane: The projective plane is what you get by gluing a disk onto itself. This object cannot be embedded in R3. Boy's surface is an immersion of this object in R3. Other surfaces, comprising "cuspidal" points, such as the Cross-cap and Steiner's Roman surface, are other representations of the projective plane in R3, which are no longer immersions, since the cuspidal points are singular points. From a transformation C "creation of cuspidal points" and its inverse C-1 "confluence of cuspidal points" it is shown how one can go from the Cross Cap to Boy's surface, via Steiner's Roman surface. Incidentally this shows how to go from a "right" Boy to a "left" Boy. It is also indicated how to permute the cuspidal points of a Cross-Cap. (In French: pages 1, 13, 14, 15 and 16)
3 - Virtual Reality: Have you ever dreamed of twirling a Steiner surface, a Moebius strip or a Boy's surface between your fingers at will? If so, first download Cosmoplayer, as freeware, then enjoy.
4 - polyhedral version of the transformation of a Cross cap into a Boy's Surface, right or left (your choice)Click here
5 -Polyhedral version of the central model of the sphere eversion.Click here.
Projects:
J.P.Petit: The eversions of the sphere and the torus, full of animated gifs.
J.P.Petit: The eversion of the cube (in preparation).
Original pages: maths_f.htm, maths_f.htm