Geodesic
A Cylinder of Revolution on Five Points
Journal for geometry and graphics, Tome 10 (2006) no. 2, pp. 207-213 Cet article a éte moissonné depuis la source Heldermann Verlag

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Although a general quadric surface is uniquely defined on nine linearly independent given points, special and possibly degenerate quadrics can be generated on fewer if certain constraints, implied or explicit, apply. E.g., coefficients of the implicit equation of a unique sphere may be generated with four given points and five constraint equations. The sphere is special but not degenerate. This article addresses a specific degenerate case, an arbitrary disposition of five given points so as to unambiguously define up to six cylinders of revolution upon them. An approach based on geometric constraints concerning the distance between any two points on the surface yields a sestic univariate in one of the cylinder axis direction numbers. Three linear variables are eliminated from the five initially formulated second order constraints. A cubic and quadratic intermediate pair of equations is produced. These contain the three homogeneous axial direction numbers. Projection of the given points onto any normal plane reveals that the five projected images lie on a circle.
Classification : 51N20, 51N05
Mots-clés : Algebraic geometry, cylinder of revolution, degenerate quadric 
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     title = {A {Cylinder} of {Revolution} on {Five} {Points}},
     journal = {Journal for geometry and graphics},
     pages = {207--213},
     year = {2006},
     volume = {10},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JGG_2006_10_2_JGG_2006_10_2_a8/}
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P. Zsombor-Murray; S. El Fashny . A Cylinder of Revolution on Five Points. Journal for geometry and graphics, Tome 10 (2006) no. 2, pp. 207-213. http://geodesic.mathdoc.fr/item/JGG_2006_10_2_JGG_2006_10_2_a8/