Altitudes of a Tetrahedron and Traceless Quadratic Forms

TL;DR

This paper explores the properties of altitudes in tetrahedra, revealing that unlike triangles, they are generally skew lines forming an equilateral hyperboloid, thus challenging the expectation of a common orthocenter.

Contribution

It provides a geometric analysis connecting tetrahedral altitudes to equilateral hyperboloids, highlighting their skew nature and contrasting with the triangle case.

Findings

Tetrahedral altitudes are generally skew lines.

Altitudes of a tetrahedron lie on an equilateral hyperboloid.

Contrasts with the concurrent altitudes of triangles.

Abstract

It is well known that the three altitudes of a triangle are concurrent at the so-called orthocenter of the triangle. So one might expect that the altitudes of a tetrahedron also meet at a point. However, it was already pointed out in 1827 by the Swiss geometer Jakob Steiner (1796--1863) that the altitudes of a general tetrahedron are mutually skew, for they are generators of an equilateral hyperboloid.