{
  "id": "1005.1033",
  "version": "v2",
  "published": "2010-05-06T16:26:45.000Z",
  "updated": "2015-12-17T15:29:03.000Z",
  "title": "Random Gaussian Tetrahedra",
  "authors": [
    "Steven Finch"
  ],
  "comment": "18 pages",
  "categories": [
    "math.PR",
    "math.MG"
  ],
  "abstract": "Given independent normally distributed points A,B,C,D in Euclidean 3-space, let Q denote the plane determined by A,B,C and D^ denote the orthogonal projection of D onto Q. The probability that the tetrahedron ABCD is acute remains intractible. We make some small progress in resolving this issue. Let Gamma denote the convex cone in Q containing all linear combinations A+r*(B-A)+s*(C-A) for nonnegative r, s. We compute the probability that D^ falls in (B+C)-Gamma to be 0.681..., but the probability that D^ falls in Gamma to be 0.683.... The intersection of these two cones is a parallelogram in Q twice the area of the triangle ABC. Among other issues, we mention the distribution of random solid angles and sums of these.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-05-06T16:26:45.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-12-17T15:29:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "51M04",
      "51M25",
      "62H10",
      "62E15",
      "52B10",
      "52A38"
    ],
    "keywords": [
      "random gaussian tetrahedra",
      "probability",
      "random solid angles",
      "gamma denote",
      "acute remains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.1033F"
    }
  }
}