Line Geometry, Sphere Geometry, Kinematics
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A Dupin cyclide is a quartic and cyclic surface. It is the envelope of a one parameter family of spheres. In Lie’s model of sphere geometry, it is represented by a conic. Lie’s line-sphere-mapping maps a conic in Lie’s quadric to a conic on Plücker’s quadric which corresponds to a regulus in the manifold of lines. Each regulus defines a ruled quadric, for example, a one-sheeted hyperboloid. Consequently, up to Lie’s line-sphere-mapping, there is no difference between a Dupin cyclide and a ruled quadric.
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