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By Steven Finch


Let S denote the unit sphere in Euclidean 3-space. A spherical triangle T is a region enclosed by three great circles on S; a great circle is a circle whose center is at the origin. The sides of T are arcs of great circles and have length a, b, c. Eachof these is ≤ π. The angle α opposite side a is the dihedral angle between the two planes passing through the origin and determined by arcs b, c. The angles β, γ opposite sides b, c are similarly defined. Each of these is ≤ π too [1]. Thesumoftheanglesis ≤ 3π yet ≥ π. In particular, the sum need not be the constant π. Define the spherical excess E = α + β + γ − π. The sum of the sides is ≥ 0yet ≤ 2π. Define the spherical defect D =2π − (a + b + c). It can be shown that the area of T is E and a calculus-based proof appears in [2]; see also [3]. Clearly the perimeter of T is 2π − D. The probability density functions for sides, angles, excess and defect on S will occupy us in this essay. Random triangles are defined here by selecting three independent uniformly distributed points on the sphere to be vertices. One way to d

Year: 2010
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