Hermitian trigonometry

Abstract

The purpose of this chapter is to survey Hermitian trigonometry, a subject that was studied at different times by several authors. These authors introduced invariants for triangles in the complex projective plane and, by different methods, they established the trigonometric laws which turn out to be two sine laws and a cosine law. In this chapter, we present and compare these laws. We shall also see that those of usual spherical trigonometry can be easily deduced from them. It turns out that one of the sine laws does not appear in the works of Blaschke and Terheggen who were the first to study Hermitian trigonometry. We show that their method directly leads to this law. In addition, we define a new triangle invariant $\alpha$ and show that this invariant and two other essential invariants defined by Blaschke and Terheggen and denoted by $\omega$ and $\Omega$ satisfy a simple equation: $\Omega =-\alpha -2\omega +(2k+1)\pi$, $k\in \mathbb{Z}$.