Random Triangles II
نویسنده
چکیده
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. Each of 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]. The sum of the angles is ≤ 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 ≥ 0 yet ≤ 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 do this is to let X1, X2, X3, Y1, Y2, Y3, Z1, Z2, Z3 be independent normally distributed random variables with mean 0 and variance 1; then the points
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