Angles as probabilities
| dc.creator | Feldman, David V. | |
| dc.creator | Klain, Daniel A. | |
| dc.date | 2008-09-19 | |
| dc.date.accessioned | 2026-07-07T10:04:15Z | |
| dc.date.available | 2026-07-07T10:04:15Z | |
| dc.description | 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. | |
| dc.description | 4 pages | |
| dc.identifier | https://arxiv.org/abs/0809.3459 | |
| dc.identifier | http://arxiv.org/abs/0809.3459 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/169599 | |
| dc.subject | Metric Geometry | |
| dc.subject | 52B11 | |
| dc.title | Angles as probabilities | |
| dc.type | text |