Construction of a Nine-Point Quadric Surface

V. A. Korotkiy

Journal for geometry and graphics, Tome 22 (2018) no. 2, pp. 183-193

Résumé

The fundamental issue of constructing a nine-point quadric was frequently discussed by mathematicians in the 19th century. They failed to find a simple linear geometric dependence that would join ten points of a quadric (similar to Pascal's theorem, which joins six points of a conic section). Nevertheless, they found different algorithms for a geometrically accurate construction (using straightedge and compass or even using straightedge alone) of a quadric that passes through nine given points. While the algorithms are quite complex, they can be implemented only with the help of computer graphics. The paper proposes a simplified computer-based realization of J. H. Engel's well-known algorithm, which makes it possible to determine the nine-point quadric and its axes of symmetry. The proposed graphics algorithm can be considered an alternative to the algebraic solution of the stated problem.

Détail

Classification : 51M15, 51M35, 51N20
Mots-clés : Biquadratic curves, pencil of quadrics, pencil of conic sections, spatial configuration of Desargues, geometrically accurate construction, computer graphics