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Journal for Geometry and Graphics 24 (2020), No. 2, 207--216
Copyright Heldermann Verlag 2020

A Rational Trigonometric Relationship Between the Dihedral Angles of a Tetrahedron and its Circumradius

Gennady Arshad Notowidigdo
School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

This paper will extend a known relationship between the circumradius and dihedral angles of a tetrahedron in three-dimensional Euclidean space to three-dimensional affine space over a general field not of characteristic two or three, using only the framework of rational trigonometry devised by Wildberger. In this framework, a linear algebraic view of trigonometry is presented, which allows the associated three-dimensional vector space of such a three-dimensional affine space to be equipped with a non-degenerate symmetric bilinear form. This will also generalise the results presented to arbitrary geometries parameterised by such a non-degenerate symmetric bilinear form.

Keywords: Rational trigonometry, tetrahedron, circumradius, dihedral angle, symmetric bilinear form.

MSC: 51M04; 51N10, 15A63, 11E04

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