Geodesic
Quadrics of Revolution on Given Points
Journal for geometry and graphics, Tome 13 (2009) no. 2, pp. 131-144.

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 51N20, 51N35
Mots-clés : Quadric of revolution, cone of revolution, cylinder of revolution, special quadric, finite given point set, repeated eigenvalues 
@article{JGG_2009_13_2_JGG_2009_13_2_a0,
     author = {A. Gfrerrer and P. J. Zsombor-Murray },
     title = {Quadrics of {Revolution} on {Given} {Points}},
     journal = {Journal for geometry and graphics},
     pages = {131--144},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2009},
     url = {http://geodesic.mathdoc.fr/item/JGG_2009_13_2_JGG_2009_13_2_a0/}
}
TY  - JOUR
AU  - A. Gfrerrer
AU  - P. J. Zsombor-Murray 
TI  - Quadrics of Revolution on Given Points
JO  - Journal for geometry and graphics
PY  - 2009
SP  - 131
EP  - 144
VL  - 13
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JGG_2009_13_2_JGG_2009_13_2_a0/
ID  - JGG_2009_13_2_JGG_2009_13_2_a0
ER  - 
%0 Journal Article
%A A. Gfrerrer
%A P. J. Zsombor-Murray 
%T Quadrics of Revolution on Given Points
%J Journal for geometry and graphics
%D 2009
%P 131-144
%V 13
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JGG_2009_13_2_JGG_2009_13_2_a0/
%F JGG_2009_13_2_JGG_2009_13_2_a0
A. Gfrerrer; P. J. Zsombor-Murray . Quadrics of Revolution on Given Points. Journal for geometry and graphics, Tome 13 (2009) no. 2, pp. 131-144. http://geodesic.mathdoc.fr/item/JGG_2009_13_2_JGG_2009_13_2_a0/