Abstract
In this chapter, we explore some of the finer points of hyperbolic geometry. We first describe the notion of convexity and explore convex sets, including the class of hyperbolic polygons. Restricting our attention to hyperbolic polygons, we go on to discuss the measurement of hyperbolic area, including the Gauss-Bonnet formula, which gives a formula for the hyperbolic area of a hyperbolic polygon in terms of its angles. We go on to use the Gauss-Bonnet formula to show that non-trivial dilations of the hyperbolic plane do not exist. We close the chapter with a discussion of the laws of trigonometry in the hyperbolic plane.
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© 1999 Springer-Verlag London
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Anderson, J.W. (1999). Convexity, Area, and Trigonometry. In: Hyperbolic Geometry. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-4471-3987-4_5
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DOI: https://doi.org/10.1007/978-1-4471-3987-4_5
Publisher Name: Springer, London
Print ISBN: 978-1-85233-156-6
Online ISBN: 978-1-4471-3987-4
eBook Packages: Springer Book Archive