On tangents to quadric surfaces

Ciprian Borcea; Xavier Goaoc; Sylvain Lazard; Sylvain Petitjean

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On tangents to quadric surfaces

Ciprian Borcea·Xavier Goaoc·Sylvain Lazard·Sylvain Petitjean

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Abstract

We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space defined 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 three-space, 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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BibTeX RIS

Ciprian Borcea, Xavier Goaoc, Sylvain Lazard, Sylvain Petitjean. 2004-02-24. On tangents to quadric surfaces. https://arxiv.org/abs/math/0402394

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