We now begin the study of hyperbolic geometry. The first step is to define a new inner product on ℝ n , 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 ℝ n+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.
KeywordsLorentz Transformation Vector Subspace Hyperbolic Geometry Geodesic Line Hyperbolic Line
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