On Tangents to Quadric Surfaces
Identifieur interne : 004D69 ( Hal/Checkpoint ); précédent : 004D68; suivant : 004D70On Tangents to Quadric Surfaces
Auteurs : Ciprian Borcea [États-Unis] ; Xavier Goaoc [France] ; Sylvain Lazard [France] ; Sylvain Petitjean [France]Source :
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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<front><div type="abstract" xml:lang="en">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?".</div>
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<abstract xml:lang="en">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?".</abstract>
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