{"id":62,"date":"2025-02-24T02:29:44","date_gmt":"2025-02-24T02:29:44","guid":{"rendered":"https:\/\/freepath.info\/?p=62"},"modified":"2025-02-24T02:34:33","modified_gmt":"2025-02-24T02:34:33","slug":"the-cauchy-formula","status":"publish","type":"post","link":"https:\/\/freepath.info\/?p=62","title":{"rendered":"the Cauchy formula"},"content":{"rendered":"

Below is a more detailed<\/strong> (yet still relatively accessible) derivation of the 3D Cauchy formula<\/strong> (also called the mean chord length formula<\/strong>) in integral geometry<\/strong>, which states:<\/p>\n

(<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>4<\/mn>\u2009<\/mtext>V<\/mi><\/mrow>A<\/mi><\/mfrac>,<\/mo><\/mrow>(L)_\\circ \\;=\\; \\frac{4\\,V}{A},<\/annotation><\/semantics><\/math><\/span><\/p>\n

where<\/p>\n

    \n
  • (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub><\/mrow>(L)_\\circ<\/annotation><\/semantics><\/math><\/span> is the mean chord length<\/em> of a convex body K<\/mi>\u2282<\/mo>R<\/mi>3<\/mn><\/msup><\/mrow>K \\subset \\mathbb{R}^3<\/annotation><\/semantics><\/math><\/span>;<\/li>\n
  • V<\/mi><\/mrow>V<\/annotation><\/semantics><\/math><\/span> is the volume<\/em> of K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span>;<\/li>\n
  • A<\/mi><\/mrow>A<\/annotation><\/semantics><\/math><\/span> is the surface area<\/em> of K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span>.<\/li>\n<\/ul>\n

    This formula can be found in classical references such as:<\/p>\n

      \n
    • L. A. Santal\u00f3, Integral Geometry and Geometric Probability<\/em> (Addison-Wesley, 1976).<\/li>\n
    • D. G. Kendall & P. A. P. Moran, Geometrical Probability<\/em> (Hafner, 1963).<\/li>\n<\/ul>\n

      Below, we give a step-by-step outline that highlights the key ideas and how one arrives at the factor 4<\/mn><\/mrow>4<\/annotation><\/semantics><\/math><\/span>.<\/p>\n
      \n

      1. The Setting: Random Chords in a Convex Body<\/h2>\n

      Let K<\/mi>\u2282<\/mo>R<\/mi>3<\/mn><\/msup><\/mrow>K \\subset \\mathbb{R}^3<\/annotation><\/semantics><\/math><\/span> be a convex body<\/em>\u2014that is, a compact, convex set with nonempty interior. We want to define a \u201crandom chord\u201d and then compute the expected length<\/em> of such a chord.<\/p>\n

      1.1 How Do We Choose a \u201cRandom Chord\u201d?<\/h3>\n

      There are several equivalent ways to define a random chord<\/em> in K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span>. The classical approach in integral geometry is:<\/p>\n
        \n
      1. Choose a random direction<\/strong> u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span> uniformly on the unit sphere S<\/mi>2<\/mn><\/msup><\/mrow>S^2<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
      2. Choose a random parallel chord<\/strong> in that direction, with a uniform distribution over all parallel chords (i.e., over all lines in direction u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span> that intersect K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span>).<\/li>\n<\/ol>\n

        Concretely, if u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span> is fixed, consider the family of planes perpendicular to u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span>. You pick one such plane \u201cat random\u201d among those that intersect K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span>. The line of intersection with K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span> (in direction u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span>) is then your random chord.<\/p>\n

        1.2 The Mean Chord Length (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub><\/mrow>(L)_\\circ<\/annotation><\/semantics><\/math><\/span><\/h3>\n

        Once we have a chord, let us denote its length by \u2113<\/mi><\/mrow>\\ell<\/annotation><\/semantics><\/math><\/span>. By mean chord length<\/em> we mean the expectation<\/p>\n

        (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>E<\/mi>[<\/mo>\u2113<\/mi>]<\/mo>.<\/mi><\/mrow>(L)_\\circ \\;=\\; \\mathbb{E}[\\ell].<\/annotation><\/semantics><\/math><\/span><\/p>\n

        We will show<\/p>\n

        (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>4<\/mn>\u2009<\/mtext>V<\/mi><\/mrow>A<\/mi><\/mfrac>.<\/mi><\/mrow>(L)_\\circ \\;=\\; \\frac{4\\,V}{A}.<\/annotation><\/semantics><\/math><\/span><\/p>\n
        \n

        2. Ingredients from Integral Geometry<\/h2>\n

        The rigorous proof relies on some foundational results of integral geometry (particularly Crofton-type<\/em> and Cauchy-type<\/em> formulas). We focus on the key geometric steps<\/strong>, omitting some of the heavier measure-theoretic justifications.<\/p>\n

        2.1 Chords, Slices, and Volume<\/h3>\n

        Let us fix a direction u<\/mi>\u2208<\/mo>S<\/mi>2<\/mn><\/msup><\/mrow>\\mathbf{u}\\in S^2<\/annotation><\/semantics><\/math><\/span>. For any real number t<\/mi><\/mrow>t<\/annotation><\/semantics><\/math><\/span>, consider the plane<\/p>\n

        P<\/mi>(<\/mo>t<\/mi>,<\/mo>u<\/mi>)<\/mo>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>{<\/mo>\u2009<\/mtext>x<\/mi>\u2208<\/mo>R<\/mi>3<\/mn><\/msup>:<\/mo>\u27e8<\/mo>x<\/mi>,<\/mo>u<\/mi>\u27e9<\/mo>=<\/mo>t<\/mi>}<\/mo>,<\/mo><\/mrow>P(t,\\mathbf{u}) \\;=\\; \\{\\,x \\in \\mathbb{R}^3 : \\langle x, \\mathbf{u}\\rangle = t\\},<\/annotation><\/semantics><\/math><\/span><\/p>\n

        where \u27e8<\/mo>\u22c5<\/mo>,<\/mo>\u22c5<\/mo>\u27e9<\/mo><\/mrow>\\langle \\cdot,\\cdot\\rangle<\/annotation><\/semantics><\/math><\/span> is the usual dot product. The intersection<\/p>\n

        K<\/mi>t<\/mi>,<\/mo>u<\/mi><\/mrow><\/msub>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>K<\/mi>\u2009<\/mtext>\u2229<\/mo>\u2009<\/mtext>P<\/mi>(<\/mo>t<\/mi>,<\/mo>u<\/mi>)<\/mo><\/mrow>K_{t,\\mathbf{u}} \\;=\\; K \\,\\cap\\, P(t,\\mathbf{u})<\/annotation><\/semantics><\/math><\/span><\/p>\n

        is a (possibly empty) cross-sectional slice<\/em> of K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span>. Then:<\/p>\n
          \n
        1. The volume<\/strong> V<\/mi><\/mrow>V<\/annotation><\/semantics><\/math><\/span> of K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span> can be recovered by integrating the areas<\/em> of the slices:\n

          V<\/mi>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>\u222b<\/mo>\u2212<\/mo>\u221e<\/mi><\/mrow>\u221e<\/mi><\/msubsup>[<\/mo>A<\/mi>r<\/mi>e<\/mi>a<\/mi><\/mrow>\u2009<\/mtext>(<\/mo>K<\/mi>t<\/mi>,<\/mo>u<\/mi><\/mrow><\/msub>)<\/mo>]<\/mo>\u2009<\/mtext>d<\/mi>t<\/mi>.<\/mi><\/mrow>V \\;=\\; \\int_{-\\infty}^{\\infty} \\bigl[\\mathrm{Area}\\,\\bigl(K_{t,\\mathbf{u}}\\bigr)\\bigr] \\,dt.<\/annotation><\/semantics><\/math><\/span><\/li>\n
        2. Chords<\/strong> parallel to u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span> arise as intersections of K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span> with lines L<\/mi>\u2225<\/mi>u<\/mi><\/mrow>L\\|\\mathbf{u}<\/annotation><\/semantics><\/math><\/span>. The set of all such lines can be parametrized by (<\/mo>t<\/mi>,<\/mo>y<\/mi>)<\/mo><\/mrow>(t,y)<\/annotation><\/semantics><\/math><\/span> where t<\/mi><\/mrow>t<\/annotation><\/semantics><\/math><\/span> is as above and y<\/mi><\/mrow>y<\/annotation><\/semantics><\/math><\/span> is a point in the plane P<\/mi>(<\/mo>t<\/mi>,<\/mo>u<\/mi>)<\/mo><\/mrow>P(t,\\mathbf{u})<\/annotation><\/semantics><\/math><\/span>. In other words, each chord is determined by picking which plane it lies in (via t<\/mi><\/mrow>t<\/annotation><\/semantics><\/math><\/span>) and where in that plane it goes through (via y<\/mi><\/mrow>y<\/annotation><\/semantics><\/math><\/span>).<\/li>\n<\/ol>\n

          2.2 Relating Chord Lengths to Surface Area<\/h3>\n

          Although volume slicing explains how chord lengths integrate to give volume<\/em>, we also need a relation to the surface area<\/em> A<\/mi><\/mrow>A<\/annotation><\/semantics><\/math><\/span>. A central insight from integral geometry is that, when averaged over all directions<\/em>, the measure of chords \u201ctouching\u201d or \u201cpiercing\u201d the boundary is proportional to A<\/mi><\/mrow>A<\/annotation><\/semantics><\/math><\/span>.<\/p>\n

          In fact, there is a known relationship sometimes called the Cauchy\u2013Crofton-type formula<\/strong> in 3D:<\/p>\n

            \n
          • The total \u201cmeasure\u201d of all lines (in all directions) that intersect K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span> is related to V<\/mi><\/mrow>V<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
          • The total \u201cmeasure\u201d of all lines (in all directions) that intersect the boundary<\/em> \u2202<\/mi>K<\/mi><\/mrow>\\partial K<\/annotation><\/semantics><\/math><\/span> is related to A<\/mi><\/mrow>A<\/annotation><\/semantics><\/math><\/span>.<\/li>\n<\/ul>\n

            One then exploits the interplay between these two measures (volume-based and area-based) to deduce the average chord length<\/em> must be a constant multiple of V<\/mi>A<\/mi><\/mfrac><\/mrow>\\tfrac{V}{A}<\/annotation><\/semantics><\/math><\/span>. The crux is pinning down that constant<\/em> as exactly 4<\/mn><\/mrow>4<\/annotation><\/semantics><\/math><\/span>.<\/p>\n
            \n

            3. Sketch of the Proof<\/h2>\n

            We will outline a reasonably direct proof, following the spirit of \u201cslicing\u201d and \u201caveraging\u201d over directions, showing why the formula must be<\/p>\n

            (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>4<\/mn>\u2009<\/mtext>V<\/mi><\/mrow>A<\/mi><\/mfrac>.<\/mi><\/mrow>(L)_\\circ \\;=\\; \\frac{4\\,V}{A}.<\/annotation><\/semantics><\/math><\/span><\/p>\n

            3.1 Counting \u201cChord-Length \u00d7<\/mo><\/mrow>\\times<\/annotation><\/semantics><\/math><\/span> Direction\u201d Two Ways<\/h3>\n
              \n
            1. Pick a Direction u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span><\/strong>
              \nLet u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span> be fixed for the moment. Consider all chords<\/em> of K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span> parallel to u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span>. You can parametrize each chord by its midpoint m<\/mi><\/mrow>m<\/annotation><\/semantics><\/math><\/span> in the plane P<\/mi>(<\/mo>t<\/mi>,<\/mo>u<\/mi>)<\/mo><\/mrow>P(t,\\mathbf{u})<\/annotation><\/semantics><\/math><\/span>. Geometrically, as m<\/mi><\/mrow>m<\/annotation><\/semantics><\/math><\/span> sweeps out the cross-section K<\/mi>t<\/mi>,<\/mo>u<\/mi><\/mrow><\/msub><\/mrow>K_{t,\\mathbf{u}}<\/annotation><\/semantics><\/math><\/span>, the corresponding chord is the line in direction u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span> through m<\/mi><\/mrow>m<\/annotation><\/semantics><\/math><\/span>, truncated by the boundary of K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span>. Its length is \u2113<\/mi>(<\/mo>m<\/mi>,<\/mo>u<\/mi>)<\/mo><\/mrow>\\ell\\bigl(m,\\mathbf{u}\\bigr)<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
            2. Integrate the Chord Length Over the Cross-Section<\/strong>
              \nIf you \u201csum up\u201d (i.e. integrate) \u2113<\/mi>(<\/mo>m<\/mi>,<\/mo>u<\/mi>)<\/mo><\/mrow>\\ell(m,\\mathbf{u})<\/annotation><\/semantics><\/math><\/span> over all midpoints m<\/mi>\u2208<\/mo>K<\/mi>t<\/mi>,<\/mo>u<\/mi><\/mrow><\/msub><\/mrow>m\\in K_{t,\\mathbf{u}}<\/annotation><\/semantics><\/math><\/span>, and then integrate over all t<\/mi><\/mrow>t<\/annotation><\/semantics><\/math><\/span>, you effectively recover a quantity proportional to the volume<\/em> V<\/mi><\/mrow>V<\/annotation><\/semantics><\/math><\/span>. (This is akin to slicing arguments in integral geometry: each small element of volume can be thought of as \u201ccontributing\u201d to exactly one chord in each direction, in measure-theoretic sense.)<\/li>\n
            3. Average Over All Directions<\/strong>
              \nAfter performing the above step for one fixed u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span>, we then let u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span> vary uniformly over the sphere S<\/mi>2<\/mn><\/msup><\/mrow>S^2<\/annotation><\/semantics><\/math><\/span>. This \u201cdouble integration\u201d (over directions u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span> and over chord-midpoints in each cross-section) can be reorganized to show a proportionality between the total chord length measure<\/em> and the ratio V<\/mi>A<\/mi><\/mfrac><\/mrow>\\tfrac{V}{A}<\/annotation><\/semantics><\/math><\/span>. The boundary \u2202<\/mi>K<\/mi><\/mrow>\\partial K<\/annotation><\/semantics><\/math><\/span> appears naturally when one keeps track of how chords terminate on \u2202<\/mi>K<\/mi><\/mrow>\\partial K<\/annotation><\/semantics><\/math><\/span>.<\/li>\n<\/ol>\n

              3.2 Pinning Down the Constant = 4<\/h3>\n

              What remains is to show that the constant<\/em> of proportionality is 4<\/strong>, rather than some other number. One way to see why is to look at special shapes<\/strong> for which the formula can be computed independently\u2014for instance, a sphere<\/strong>.<\/p>\n

              Example: The Unit Sphere<\/h4>\n

              Consider K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span> as the unit sphere<\/em> in R<\/mi>3<\/mn><\/msup><\/mrow>\\mathbb{R}^3<\/annotation><\/semantics><\/math><\/span>. Then:<\/p>\n
                \n
              • V<\/mi>=<\/mo>4<\/mn>3<\/mn><\/mfrac>\u03c0<\/mi>r<\/mi>3<\/mn><\/msup><\/mrow>V = \\tfrac{4}{3}\\pi r^3<\/annotation><\/semantics><\/math><\/span> with r<\/mi>=<\/mo>1<\/mn><\/mrow>r=1<\/annotation><\/semantics><\/math><\/span>, so V<\/mi>=<\/mo>4<\/mn>\u03c0<\/mi><\/mrow>3<\/mn><\/mfrac><\/mrow>V = \\tfrac{4\\pi}{3}<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
              • A<\/mi>=<\/mo>4<\/mn>\u03c0<\/mi>r<\/mi>2<\/mn><\/msup><\/mrow>A = 4\\pi r^2<\/annotation><\/semantics><\/math><\/span> with r<\/mi>=<\/mo>1<\/mn><\/mrow>r=1<\/annotation><\/semantics><\/math><\/span>, so A<\/mi>=<\/mo>4<\/mn>\u03c0<\/mi><\/mrow>A = 4\\pi<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
              • Hence, V<\/mi>A<\/mi><\/mfrac>=<\/mo>4<\/mn>\u03c0<\/mi>\/<\/mi>3<\/mn><\/mrow>4<\/mn>\u03c0<\/mi><\/mrow><\/mfrac>=<\/mo>1<\/mn>3<\/mn><\/mfrac><\/mrow>\\tfrac{V}{A} = \\tfrac{4\\pi\/3}{4\\pi} = \\tfrac{1}{3}<\/annotation><\/semantics><\/math><\/span>.<\/li>\n<\/ul>\n

                Now, what is the mean chord length<\/strong> (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub><\/mrow>(L)_\\circ<\/annotation><\/semantics><\/math><\/span> of a unit sphere<\/em> if chords are chosen in the manner above?<\/p>\n
                  \n
                • A random direction u<\/mi><\/mrow>\\mathbf{u}<\/annotation><\/semantics><\/math><\/span> is irrelevant for a sphere (all directions are \u201cthe same\u201d), so we just pick a random chord uniformly among all parallel chords.<\/li>\n
                • For a sphere, all parallel chords<\/em> are parallel \u201cslices.\u201d If you pick a random chord by uniformly picking its midpoint inside the circular cross-section, classical geometric probability shows that the midpoint is more likely<\/em> to be near the center. The resulting mean chord length (in a unit sphere) turns out to be 4<\/mn>3<\/mn><\/mfrac>r<\/mi>=<\/mo>4<\/mn>3<\/mn><\/mfrac><\/mrow>\\tfrac{4}{3}r = \\tfrac{4}{3}<\/annotation><\/semantics><\/math><\/span> when r<\/mi>=<\/mo>1<\/mn><\/mrow>r=1<\/annotation><\/semantics><\/math><\/span>.<\/li>\n<\/ul>\n

                  In other words, one can do a quick integral or use well-known results about chord distributions in a circle\/sphere to see that the expected chord length in a unit sphere<\/em> is 4<\/mn>3<\/mn><\/mfrac><\/mrow>\\tfrac{4}{3}<\/annotation><\/semantics><\/math><\/span>. Comparing this with V<\/mi>A<\/mi><\/mfrac>=<\/mo>1<\/mn>3<\/mn><\/mfrac><\/mrow>\\tfrac{V}{A} = \\tfrac{1}{3}<\/annotation><\/semantics><\/math><\/span> for r<\/mi>=<\/mo>1<\/mn><\/mrow>r=1<\/annotation><\/semantics><\/math><\/span>, we get:<\/p>\n

                  (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>4<\/mn>\u00d7<\/mo>V<\/mi>A<\/mi><\/mfrac>,<\/mo><\/mrow>(L)_\\circ \\;=\\; 4 \\times \\frac{V}{A},<\/annotation><\/semantics><\/math><\/span><\/p>\n

                  confirming the constant must indeed be 4<\/mn><\/mrow>4<\/annotation><\/semantics><\/math><\/span>.<\/p>\n

                  For any<\/em> convex body K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span>, a more sophisticated integral-geometric argument (a \u201ctransport-of-measure\u201d or \u201cCrofton\u2013Santal\u00f3 approach\u201d) generalizes exactly this result to show the constant remains 4<\/mn><\/mrow>4<\/annotation><\/semantics><\/math><\/span>.<\/p>\n
                  \n

                  4. Making It More Rigorous: Crofton\u2013Santal\u00f3 Formalism<\/h2>\n

                  A more rigorous treatment involves:<\/p>\n

                    \n
                  1. Measure of Lines in 3D<\/strong>
                    \nOne defines a measure on the space of all lines in R<\/mi>3<\/mn><\/msup><\/mrow>\\mathbb{R}^3<\/annotation><\/semantics><\/math><\/span> that is invariant<\/em> under rigid motions (translations + rotations). Concretely, this measure can be described by choosing u<\/mi>\u2208<\/mo>S<\/mi>2<\/mn><\/msup><\/mrow>\\mathbf{u}\\in S^2<\/annotation><\/semantics><\/math><\/span> (the direction) uniformly and then choosing the perpendicular distance from the origin in a certain way.<\/li>\n
                  2. Counting the Lines That Intersect K<\/mi><\/mrow>K<\/annotation><\/semantics><\/math><\/span><\/strong>
                    \nThe integral of the \u201cindicator function\u201d 1<\/mn>{<\/mo>line<\/mtext>\u2229<\/mo>K<\/mi>\u2260<\/mo>\u2205<\/mi>}<\/mo><\/mrow>\\mathbf{1}\\{\\text{line} \\cap K \\neq \\emptyset\\}<\/annotation><\/semantics><\/math><\/span> with respect to this measure on lines is proportional<\/em> to V<\/mi><\/mrow>V<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
                  3. Counting the Lines That Hit the Boundary \u2202<\/mi>K<\/mi><\/mrow>\\partial K<\/annotation><\/semantics><\/math><\/span><\/strong>
                    \nLikewise, if one counts lines according to where and how they intersect \u2202<\/mi>K<\/mi><\/mrow>\\partial K<\/annotation><\/semantics><\/math><\/span>, one obtains an expression proportional<\/em> to A<\/mi><\/mrow>A<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
                  4. Relating Total Chord Length<\/strong>
                    \nWhen you want the sum<\/em> (or integral<\/em>) of chord lengths over all such random lines, you effectively do a more refined version of (2) and (3). The difference between the boundary-based counting and volume-based counting isolates a factor that emerges as exactly 4<\/mn><\/mrow>4<\/annotation><\/semantics><\/math><\/span>.<\/li>\n<\/ol>\n

                    Putting it all together yields the final statement:<\/p>\n

                    (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>4<\/mn>\u2009<\/mtext>V<\/mi><\/mrow>A<\/mi><\/mfrac>.<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow>\\boxed{ (L)_\\circ \\;=\\; \\frac{4\\,V}{A}. }<\/annotation><\/semantics><\/math><\/span><\/p>\n
                    \n

                    5. Additional Perspective and Intuition<\/h2>\n
                      \n
                    1. Dimensional Analysis<\/strong>\n
                        \n
                      • [<\/mo>V<\/mi>]<\/mo><\/mrow>[V]<\/annotation><\/semantics><\/math><\/span> has dimensions of (length)<\/mtext>3<\/mn><\/msup><\/mrow>\\text{(length)}^3<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
                      • [<\/mo>A<\/mi>]<\/mo><\/mrow>[A]<\/annotation><\/semantics><\/math><\/span> has dimensions of (length)<\/mtext>2<\/mn><\/msup><\/mrow>\\text{(length)}^2<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
                      • Thus, V<\/mi>A<\/mi><\/mfrac><\/mrow>\\tfrac{V}{A}<\/annotation><\/semantics><\/math><\/span> has dimension of (length)<\/mtext><\/mrow>\\text{(length)}<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
                      • If there is a universal constant<\/em> relating a mean chord length<\/em> to V<\/mi><\/mrow>V<\/annotation><\/semantics><\/math><\/span> and A<\/mi><\/mrow>A<\/annotation><\/semantics><\/math><\/span>, it must be dimensionless\u2014and turns out to be 4<\/mn><\/mrow>4<\/annotation><\/semantics><\/math><\/span> in 3D.<\/li>\n<\/ul>\n<\/li>\n
                      • Analogy with 2D<\/strong>
                        \nIn 2D<\/strong>, the analogous formula says that the mean chord length of a convex domain (mean \u201crandom segment\u201d through the region) is<\/p>\n

                        (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub>\u2005\u200a<\/mtext>=<\/mo>\u2005\u200a<\/mtext>\u03c0<\/mi>\u2009<\/mtext>(<\/mo>Area<\/mtext>)<\/mo><\/mrow>Perimeter<\/mtext><\/mfrac>.<\/mi><\/mrow>(L)_\\circ \\;=\\; \\frac{\\pi\\,(\\text{Area})}{\\text{Perimeter}}.<\/annotation><\/semantics><\/math><\/span>The constant there is \u03c0<\/mi><\/mrow>\\pi<\/annotation><\/semantics><\/math><\/span>. In 3D<\/strong>, it becomes 4<\/mn><\/mrow>4<\/annotation><\/semantics><\/math><\/span>. One might have guessed a \u03c0<\/mi><\/mrow>\\pi<\/annotation><\/semantics><\/math><\/span>-related constant in 3D as well, but the integral geometry shows that 4<\/mn><\/mrow>4<\/annotation><\/semantics><\/math><\/span> emerges instead.<\/li>\n
                      • Special Cases<\/strong>\n
                          \n
                        • Sphere<\/strong> (already seen): mean chord length =<\/mo>4<\/mn>3<\/mn><\/mfrac>r<\/mi><\/mrow>= \\tfrac{4}{3}r<\/annotation><\/semantics><\/math><\/span>.<\/li>\n
                        • Cube<\/strong> or Rectangular Box<\/strong>: one can attempt a direct (though tedious) calculation or rely on the general theorem.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n
                          \n

                          6. Concluding Remarks<\/h2>\n

                          The Cauchy mean chord length formula<\/strong> in 3D,<\/p>\n

                          (<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub>=<\/mo>4<\/mn>V<\/mi><\/mrow>A<\/mi><\/mfrac><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose>,<\/mo><\/mrow>\\boxed{(L)_\\circ = \\frac{4V}{A}},<\/annotation><\/semantics><\/math><\/span><\/p>\n

                          is a cornerstone of integral geometry. It reveals a deep connection among:<\/p>\n

                            \n
                          • The 1D<\/strong> quantity: average chord length,<\/li>\n
                          • The 2D<\/strong> boundary measure: A<\/mi><\/mrow>A<\/annotation><\/semantics><\/math><\/span>,<\/li>\n
                          • The 3D<\/strong> interior measure: V<\/mi><\/mrow>V<\/annotation><\/semantics><\/math><\/span>.<\/li>\n<\/ul>\n

                            While the final statement (<\/mo>(<\/mo>L<\/mi>)<\/mo>\u2218<\/mo><\/msub>=<\/mo>4<\/mn>V<\/mi>\/<\/mi>A<\/mi>)<\/mo><\/mrow>\\bigl((L)_\\circ = 4V\/A\\bigr)<\/annotation><\/semantics><\/math><\/span> looks simple, its proof showcases the power of geometric probability and the invariance properties of \u201crandom directions\u201d and \u201crandom chords.\u201d The sphere example clinches why the constant is specifically 4<\/mn><\/mrow>4<\/annotation><\/semantics><\/math><\/span>. For a fully rigorous derivation\u2014with all measure-theoretic details\u2014one consults the classic treatments in Santal\u00f3\u2019s or Kendall & Moran\u2019s texts, which systematically develop the necessary Crofton\u2013Cauchy\u2013Santal\u00f3<\/em> formulas in multiple dimensions.<\/p>\n","protected":false},"excerpt":{"rendered":"

                            Below is a more detailed (yet still relatively accessible) derivation of the 3D Cauchy formula (also called the mean chord length formula) in integral geometry, which states: (L)\u2218\u2005\u200a=\u2005\u200a4\u2009VA,(L)_\\circ \\;=\\; \\frac{4\\,V}{A}, where (L)\u2218(L)_\\circ is the mean chord length of a convex body K\u2282R3K \\subset \\mathbb{R}^3; VV is the volume of KK; AA is the surface area […]<\/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-62","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\/62","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=62"}],"version-history":[{"count":3,"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts\/62\/revisions"}],"predecessor-version":[{"id":67,"href":"https:\/\/freepath.info\/index.php?rest_route=\/wp\/v2\/posts\/62\/revisions\/67"}],"wp:attachment":[{"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=62"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=62"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/freepath.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=62"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}