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- W2953028112 abstract "We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a tetrahedron T, divided by 2*pi, gives the probability that an orthogonal projection of T onto a random 2-plane is a triangle. More generally, it is shown that the sum of the (solid) inner vertex angles of an n-simplex S, normalized by the area of the unit (n-1)-hemisphere, gives the probability that an orthogonal projection of S onto a random hyperplane is an (n-1)-simplex. Applications to more general polytopes are treated briefly, as is the related Perles-Shephard proof of the classical Gram-Euler relations." @default.
- W2953028112 created "2019-06-27" @default.
- W2953028112 creator A5053495835 @default.
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- W2953028112 date "2008-09-19" @default.
- W2953028112 modified "2023-09-27" @default.
- W2953028112 title "Angles as probabilities" @default.
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- W2953028112 hasPublicationYear "2008" @default.
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