{"id":107,"date":"2025-03-17T01:35:26","date_gmt":"2025-03-17T01:35:26","guid":{"rendered":"https:\/\/freepath.info\/?p=107"},"modified":"2025-03-17T01:35:26","modified_gmt":"2025-03-17T01:35:26","slug":"overview-of-common-chord-picking-methods","status":"publish","type":"post","link":"https:\/\/freepath.info\/?p=107","title":{"rendered":"Overview of Common Chord-Picking Methods"},"content":{"rendered":"

Below is an explanation of the table you provided, describing different chord distributions<\/strong> in a sphere under various sampling models (ways of selecting chords). These chord\u2010selection schemes are classic examples in geometric probability<\/strong> (especially for spheres in 3D). The table shows:<\/p>\n

    \n
  1. The model<\/strong> or method for choosing chords.<\/li>\n
  2. The mean chord length<\/strong> \"G(r)\" (or some normalized version).<\/li>\n
  3. A parameter \"\beta_1\" related to the mean or distribution.<\/li>\n
  4. The variance<\/strong> of chord length (\"\mathrm{Var}(l)\").<\/li>\n<\/ol>\n

    These items relate to how one picks \u201crandom chords\u201d in a sphere and how this affects the resulting probability distribution of chord lengths.<\/p>\n

    \n

    1. Overview of Common Chord-Picking Methods<\/h2>\n

    For a sphere<\/strong> (typically of unit radius for convenience), there are several classical ways to define a \u201crandom chord,\u201d each yielding a different distribution of chord lengths. The most famous is the \u201cBertrand paradox\u201d in 2D (circle chords), which shows that \u201crandom chord\u201d is ambiguous unless one specifies how<\/em> the chord is chosen. In 3D, analogous ambiguities arise when choosing \u201crandom chords\u201d in a sphere. Below are some well-known sampling methods (line type \"L[;]\" in your notation, i.e., surface\u2013surface chords):<\/p>\n

      \n
    1. Method 1<\/strong>: Pick two random points independently on the surface (uniformly on the surface), then join them.<\/li>\n
    2. Method 3<\/strong>: Pick a chord by choosing its midpoint at a uniform distance from the center (i.e., a \u201cspherical shell\u201d approach in 3D).<\/li>\n
    3. Method 4<\/strong>: Pick the chord by choosing its midpoint uniformly in the volume<\/strong> of the sphere, then a random direction through that midpoint.<\/li>\n
    4. Method 8<\/strong>: Some specific arrangement where the chord is normal to a plane of a random great circle, etc. (It often resembles \u201crandom orientation, then random point in the circle cross-section,\u201d leading to another standard distribution.)<\/li>\n<\/ol>\n

      Each of these corresponds to a different notion of randomness and thus yields a different chord\u2010length distribution.<\/p>\n

      \n

      2. Table Columns<\/h2>\n

      From the excerpt:<\/p>\n\n\n\n\n\n\n\n\n
      #<\/th>\nModel (Line Type \"L[;]\")<\/th>\n\"G(r)\"<\/th>\n\"\beta_1\"<\/th>\nVariance(\"l\")<\/th>\n<\/tr>\n<\/thead>\n
      1<\/td>\nchord joining 2 random points on sphere surface<\/td>\n\"\frac{\mathfrak{L}}{2}\"<\/td>\n4\/3<\/td>\n2\/9<\/td>\n<\/tr>\n
      3<\/td>\nchord center uniformly distributed distance from center<\/td>\n\"\frac{\mathfrak{L}}{2}<\/td>\n\"\pi\/2\"<\/td>\n0.199<\/td>\n<\/tr>\n
      4<\/td>\nchord center uniformly distributed inside sphere, random direction<\/td>\n\"\tfrac{3}{8},\mathfrak{L}<\/td>\n\"3\pi\/8\"<\/td>\n0.212<\/td>\n<\/tr>\n
      8<\/td>\nnormal to plane of random great circle, etc.<\/td>\n\"\frac{\mathfrak{L}}{2}\"<\/td>\n4\/3<\/td>\n2\/9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Here:<\/p>\n

        \n
      • \"\mathfrak{L}\"<\/strong> usually denotes the diameter<\/em> of the sphere or some reference length (sometimes 2 for a unit sphere). In many references, \"\mathfrak{L}=2R\", with \"R=1\" as the sphere\u2019s unit radius, so \"\mathfrak{L}=2\". Then chord length \"l\" ranges from 0 to \"\mathfrak{L}=2\" (for a unit sphere).<\/li>\n
      • \"G(r)\"<\/strong> might be the mean chord length<\/em> (or an expression used in the distribution function). The notation can vary, but typically \"G(r)\" could be the expected chord length or a function of chord midpoint position.<\/li>\n
      • \"\beta_1\"<\/strong> is presumably a dimensionless factor related to the mean chord length or some ratio of integrals. Often in geometric probability, one writes an expected chord length as \"\langle. So maybe \"\beta_1. In the table, for some methods, \"\beta_1. This is a known result: if two random points are chosen on a sphere of diameter \"\mathfrak{L}=2\", the mean chord length is \"\langle. So \"\beta_1=4\/3\" there.<\/li>\n
      • Variance(\"l\")<\/strong> is the variance of chord length under that random-chord model.<\/li>\n<\/ul>\n
        \n

        Method 1 & Method 8: \"\beta_1<\/h3>\n

        For a unit sphere<\/strong>, the mean chord length if endpoints are chosen uniformly on the surface is \"\tfrac{4R}{3}=\tfrac{4}{3}\" because \"R=1\". If \"\mathfrak{L}=2\" (the diameter for a unit sphere), then \"\langle, and thus \"\beta_1=4\/3\". The table shows that for cases 1<\/strong> and 8<\/strong>, \"\beta_1=4\/3\". Indeed, for certain symmetrical ways of sampling chords in the sphere, the chord-length distribution ends up identical, giving the same mean and variance. This matches well-known results in geometric probability.<\/p>\n

        \n

        3. Interpreting the Expressions in the Table<\/h2>\n
          \n
        • Chord length distribution<\/strong>: Often expressed in the form \"f(l)\" over \"0. Each model picks chords in a different manner, so the resulting PDFs differ.<\/li>\n
        • \"\langle<\/strong>: The mean chord length. This might be given in a closed form or as an integral. Sometimes the table instead shows partial expressions (like \"\tfrac{\mathfrak{L}}{2}\sqrt{4-\mathfrak{L}^2}\") or factors like \"\beta_1\" that scale with the radius \"R\".<\/li>\n
        • Variance(\"l\")<\/strong>: The second central moment minus the square of the first moment, i.e. \"\langle.<\/li>\n<\/ul>\n

          For Method 3<\/strong> (chord center uniformly distributed on a line from center) and Method 4<\/strong> (chord midpoint uniformly in the volume<\/em>), the average chord lengths turn out smaller or larger depending on the bias introduced by the sampling. In particular:<\/p>\n

            \n
          • If you pick chord midpoints near the sphere\u2019s center more frequently (Method 4) vs. near the surface, you\u2019ll get systematically longer chords on average.<\/li>\n
          • Conversely, if chord endpoints are restricted to the surface (Method 1), you get a different average chord length.<\/li>\n<\/ul>\n

            These differences produce distinct values for \"\beta_1\" and the variance. The numbers 0.199 and 0.212 likely correspond to dimensionless variances for those models\u2014these are somewhat standard results from integral geometry or direct integration.<\/p>\n

            \n

            4. Key Takeaway<\/h2>\n

            The table is summarizing known formulas<\/strong> for mean chord length (and chord-length variance) under various \u201crandom chord\u201d selection rules<\/strong> in a (presumably) 3D sphere. The references to \"\beta_1\", etc., capture how the expected chord length<\/strong> scales with the sphere radius or diameter. Some methods coincide in that they produce the same distribution (hence the same mean and variance, e.g., #1 and #8). Others (#3 and #4) produce different distributions and thus different means\/variances.<\/p>\n

            Relation to Known Equations<\/h3>\n

            All these cases are variants of the \u201cBertrand problem\u201d generalized to 3D (sometimes called the Bertrand\u2013Sylvester problem<\/strong>), where chord generation method<\/em> must be specified. The results align with standard integral geometry theorems, and the presence of \"\frac{4}{3}\" is quite typical for the average chord length in a sphere when endpoints are uniformly distributed on the surface.<\/p>\n

            \n

            References & Context<\/h3>\n
              \n
            • Solomon (1978) \u201cGeometric Probability\u201d<\/strong> discusses chord problems in 2D and 3D, chord distributions in spheres, and \u201crandom secant\u201d approaches.<\/li>\n
            • Bertrand\u2019s paradox<\/strong> (originally for circles) generalizes to spheres, leading to multiple distinct \u201crandom chord\u201d definitions.<\/li>\n
            • Integral geometry<\/strong> (e.g., Santalo, Chern, etc.) also addresses chord distributions in higher dimensions.<\/li>\n<\/ul>\n
              \n

              Final Summary<\/h2>\n

              Your table compares four chord-generation methods<\/strong> in a sphere (labeled #1, #3, #4, #8). Each method yields a different probability distribution of chord lengths<\/strong>, with different mean chord length<\/strong> (\"\beta_1\") and variance<\/strong>. Some methods produce the same result (cases #1 and #8 both give \"\beta_1), others differ (\"\beta_1, etc.). This underscores how the notion of \u201crandom chord\u201d can vary, each interpretation requiring a distinct geometric probability analysis.<\/p>\n","protected":false},"excerpt":{"rendered":"

              Below is an explanation of the table you provided, describing different chord distributions in a sphere under various sampling models (ways of selecting chords). These chord\u2010selection schemes are classic examples in geometric probability (especially for spheres in 3D). The table shows: The model or method for choosing chords. The mean chord length (or some normalized […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-107","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"brizy_media":[],"_links":{"self":[{"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts\/107","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=107"}],"version-history":[{"count":2,"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts\/107\/revisions"}],"predecessor-version":[{"id":109,"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts\/107\/revisions\/109"}],"wp:attachment":[{"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=107"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=107"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=107"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}