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Journal for Geometry and Graphics 22 (2018), No. 2, 183--193
Copyright Heldermann Verlag 2018

Construction of a Nine-Point Quadric Surface

Viktor A. Korotkiy
Inst. of Architecture and Construction, South Ural State University, 76 Lenin prospect, Chelyabinsk 454080, Russia

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.

Keywords: Biquadratic curves, pencil of quadrics, pencil of conic sections, spatial configuration of Desargues, geometrically accurate construction, computer graphics.

MSC: 51M15; 51M35, 51N20

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