Hyperbolic Geometry

Abstract

We now begin the study of hyperbolic geometry. The first step is to define a new inner product on ℝⁿ, called the Lorentzian inner product. This leads to a new concept of length. In particular, imaginary lengths are possible. In Section 3.2, hyperbolic n-space is defined to be the positive half of the sphere of unit imaginary radius in ℝⁿ⁺¹. The elements of hyperbolic arc length and volume are determined in Sections 3.3 and 3.4. The chapter ends with a section on hyperbolic trigonometry.