On Tangents to Quadric Surfaces

Ciprian Borcea 1 Xavier Goaoc 2 Sylvain Lazard 2 Sylvain Petitjean 2
2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to congurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space dened by all complex quadric surfaces which express the fact that several quadrics are tangent along a curve to one and the same quadric of rank at least three, and called, for intuitive reasons: a basket. Lines in any ruling of the latter will be common tangents. These considerations are then restricted to spheres in Euclidean threespace, and result in a complete answer to the question over the reals: "When do four spheres allow infinitely many common tangents?".
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Ciprian Borcea, Xavier Goaoc, Sylvain Lazard, Sylvain Petitjean. On Tangents to Quadric Surfaces. 2004. ⟨inria-00431701⟩

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