An Area Formula For A Hyperbolic Quadrilateral

In this thesis we obtain a formula for the area of a hyperbolic quadrilateral interms of the lengths of its sides and diagonals. In fact, a similar formula for a sphericalquadrilateral had been obtained by Casey more than a century ago. Similar to Casey’sapproach, we work in the hyperboloid model in the Lorentzian space of the hyperbolicplane. Under the Lorentzian stereographic projection from the South pole, a hyperbolicquadrilateral is projected to a circular one in the equator Euclidean plane. Sincethe Lorentzian stereographic projection preserves angles and circles, the problem istransformed to solving a trigonometric problem for a certain Euclidean triangle.The thesis is organized as follows. In Chapter1we briefly outline some basic factsabout Lorentzian space and the hyperboloid model of the hyperbolic space. In Chapter2we introduce the notion of Lorentzian inversions in Lorentzian spheres, as well as thatof Lorentzian stereographic projection, and discuss some of their properties such as theLorentzian angle-preserving. Finally, in Chapter3we state and prove an area formulaof a hyperbolic quadrilateral in terms of the lengths of its sides and diagonals.