Rectangular Hexahedrons as Fermat Bases of Quadrics
DOI:
https://doi.org/10.5644/SJM.09.2.15Keywords:
Fermat (geometric) problem, semicircle, rectangular hexahedron, quadric, sphere, ellipsoid, rotational paraboloid, hyperboloid with two sheetsAbstract
The concepts of a Fermat base of a quadric and of a Fermat locus of a rectangular hexahedron come from an old geometric problem by Pierre de Fermat about a semicircle on a side of a rectangle with ratio of adjacent sides equal to $\sqrt{2}$, which was resolved by synthetic methods first by Leonard Euler in 1750. An arbitrary rectangular hexahedron has a quadric as its Fermat locus. This quadric is either an ellipsoid, a rotational paraboloid or a hyperboloid with two sheets. Conversely, for every quadric from any of these three types one can ask to find all of its rectangular hexahedron Fermat bases which share a line of symmetry with the quadric.
2010 Mathematics Subject Classification. Primary 54H01
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