Abstract
In the pseudoeuclidean space1 E 3 a surface Δ of degree 3 or 4 is calledDupin-cyclide if there exists a (pseudoeuclidean) torus Θ such that a (pseudoeuclidean) spherical inversion maps Θ upon Δ. If the axis of Θ is respectively space-like, isotropic or time-like Δ is calledDupin-r-,-l- or-z-cyclide. ADupin-cyclide Δ is the envelope of two families of spheres, the caustic surfaces of Δ degenerate in caustic curves and the lines of curvature of Δ constitute two (orthogonal) families of circles.
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Schmidt, G. Die Dupinschen Zykliden im pseudoeuklidischen Raum. J Geom 60, 146–159 (1997). https://doi.org/10.1007/BF01252224
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DOI: https://doi.org/10.1007/BF01252224