![]() (invariant under the action of a half-turn around x = y,z = 0)Ĭf. The ring parabolic cyclide with isometric directrix circles is invariant under the action of a half-turn, since a half-turn swaps the two faces.Īfter simplifying by a change of frame, the equation of this cyclide is Ring parabolic cyclide with isometric directrix circles With on the first parabola, and on the second one. The Cartesian parametrization of the cyclide by the double family of curvature circles is:ġ/2) the radii of the directrix circles being kp/2 and (1 – The center of the inversion mapping the ring cyclide onto the torus is. The sphere perpendicular to the previous ones has center The tangent spheres are the spheres with center and radius and the spheres with center Not to be mistaken for the pinched torus. If is the current point on the ellipse and the current point on the hyperbola, then the current point on the cyclide is, hence its Cartesian parametrization by the double family of curvature circles: where d is such that the radii of the directrix circles are. The sphere with center on one of the conic, the envelope of which is the cyclide, is tangent to the cyclide at a circle, intersection between this sphere and the cone with vertex the center of the sphere and directrix the other conic.Įquations of the ellipto-hyperbolic cyclides: The horizontal circles intersect, the vertical circles are outside of one another Secant directrix circles of the plane of the ellipseĭirectrix circles of the plane of the hyperbola outside of one another We get the "horn" cyclide in the following cases: One of the horizontal circles contains the other one, the vertical circles intersect. The hyperbola is one of the equidistance curves of the two circles. The directrix circles of the plane of the hyperbola intersect. One of the directrix circles of the plane of the ellipse contains the other one and the generatrix spheres of the cyclide are tangent internally to the interior circle the ellipse is one of the We get the "spindle" cyclide in the following cases: We represented the sphere with center I 1 passing by M that is tangent to the cyclide, and on the right the sphere with center I 2. ![]() General view with the 4 directrix circles and the 2 focals.Īny normal at M to the cyclide meets the focal ellipse at I 1 and the focal hyperbola at I 2. The directrix circles of the plane of the hyperbola are outside of one another. In bold black, trace of the sphere perpendicular to the generatrix spheres (sphere of the inversion swapping the two directrix circles). One of the directrix circles of the plane of the ellipse contains the other one, and the generatrix spheres of the cyclide are externally tangent to these circles the ellipse is one of the equidistance curves of the two circles. We then get the "ring" cyclide in the following cases: The 4 circles, sections by the planes of the two conics, are called directrix circles of the cyclide (they define it completely). The lines joining two points of each focal conic are the normals of the cyclide (hence the explanation of the previous property: the previous two tangent planes are perpendicular respectively to the two curvature lines at a point of the surface since the curvature lines are orthogonal, so are the planes). or two parabolas, and the cyclide is called "parabolic". To the eyes, these conics seem to intersect with a right angle. The two conics have the property that for any pair of points I 1 and I 2 belonging to both the conics, the two tangent planes to both the conics at I 1 either an ellipse and a hyperbola, and the cyclide is called "ellipto-hyperbolic". There are two cases: the focal conics are. These conics are called the focal conics of the cyclide. It can be proved that the two focal curves (loci of the centers of the spheres) are conics located in orthogonal planes and such that the foci of one are the vertices of the other this locus constitutes the focal of the cyclide. The spheres are then tangent to the cyclide along circles, which are the curvature lines. ![]() It is conjectured that they are the only surfaces that have a double generation by circles, when the two families of circles are orthogonal.Ģ) Second definition (directly equivalent to the previous one): they are the surfaces that are envelopes of spheres, in two different ways. , p 646, p 204, p 478.įrom the Greek kuklos: circle, wheel and eidos: appearance.Ĭharles Dupin (1784-1873): French economist, mathematician and politician.ġ) First definition: the Dupin cyclides (in the strict sense) are the surfaces, different from the tori, the curvature lines of which are circles (as an exception, straight lines). Surface studied by Dupin in 1822, Darboux in 1872 and Forsyth in 1912. ![]()
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