Article Dans Une Revue Journal for Geometry and Graphics Année : 2009

Quadrics of Revolution on Given Points

Anton Gfrerrer
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Résumé

In general, 4 points define a 3 parameter set of axisymmetric quadrics while 5 and 6 given points reduce these 3 degrees of freedom to 2 and 1, respectively. Similarly, 7 supporting points confine members of the set to a finite number. By imposing 2 constraints on the quadric coefficient matrix the 5 points are sufficient to find the axis direction of up to 6 right cylinders. Imposing only 1 constraint allows 6 points to support up to 12 right cones. Without either constraint, that implies a singular coefficient matrix or singular conic submatrix, up to 4 quadrics of revolution, possibly of mixed species, can contain 7 points. Formal arguments and proofs are presented to substantiate these observations. Algorithms are developed and applied to exhibit cases with 6 right cylinders, 12 right cones and 4 quadrics of revolution, at least 3 of which are of different type. Spheres, being uniquely defined on 4 points, are specifically excluded from consideration. The cases of 12 cones and 4 quadrics of revolution are believed to be original revelations. Methods to fit quadrics of revolution to more than 7 points are suggested.

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hal-04061222 , version 1 (06-04-2023)

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Anton Gfrerrer, Paul J Zsombor-Murray. Quadrics of Revolution on Given Points. Journal for Geometry and Graphics, 2009, 13 (2), pp.131-144. ⟨hal-04061222⟩
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