Overview of Common Chord-Picking Methods

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‐selection 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 $G(r)$ (or some normalized version).
A parameter $\beta_1$ related to the mean or distribution.
The variance of chord length ( $\mathrm{Var}(l)$ ).

These items relate to how one picks “random chords” in a sphere and how this affects the resulting probability distribution of chord lengths.

1. Overview of Common Chord-Picking Methods

For a sphere (typically of unit radius for convenience), there are several classical ways to define a “random chord,” each yielding a different distribution of chord lengths. The most famous is the “Bertrand paradox” in 2D (circle chords), which shows that “random chord” is ambiguous unless one specifies how the chord is chosen. In 3D, analogous ambiguities arise when choosing “random chords” in a sphere. Below are some well-known sampling methods (line type $L[;]$ in your notation, i.e., surface–surface chords):

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

Each of these corresponds to a different notion of randomness and thus yields a different chord‐length distribution.

2. Table Columns

From the excerpt:

#	Model (Line Type $L[;]$ )	$G(r)$	$\beta_1$	Variance( $l$ )
1	chord joining 2 random points on sphere surface	$\frac{\mathfrak{L}}{2}$	4/3	2/9
3	chord center uniformly distributed distance from center	$\frac{\mathfrak{L}}{2} \sqrt{4 - \mathfrak{L}^2}$	$\pi/2$	0.199
4	chord center uniformly distributed inside sphere, random direction	$\tfrac{3}{8},\mathfrak{L} \sqrt{4 - \mathfrak{L}^2}$	$3\pi/8$	0.212
8	normal to plane of random great circle, etc.	$\frac{\mathfrak{L}}{2}$	4/3	2/9

Here:

$\mathfrak{L}$ usually denotes the diameter 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’s unit radius, so $\mathfrak{L}=2$ . Then chord length $l$ ranges from 0 to $\mathfrak{L}=2$ (for a unit sphere).
$G(r)$ might be the mean chord length (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.
$\beta_1$ 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 $\langle l \rangle = \beta \times \mathfrak{L}$ . So maybe $\beta_1 = \frac{\langle l \rangle}{\mathfrak{L}}$ . In the table, for some methods, $\beta_1 = 4/3$ . This is a known result: if two random points are chosen on a sphere of diameter $\mathfrak{L}=2$ , the mean chord length is $\langle l \rangle = \frac{4R}{3} = \left(\frac{4}{3}\right)\times \left(\frac{\mathfrak{L}}{2}\right)$ . So $\beta_1=4/3$ there.
Variance( $l$ ) is the variance of chord length under that random-chord model.

Method 1 & Method 8: $\beta_1 = 4/3$

For a unit sphere, 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 $\langle l \rangle = \tfrac{4}{3} \approx 1.333$ , and thus $\beta_1=4/3$ . The table shows that for cases 1 and 8, $\beta_1=4/3$ . 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.

3. Interpreting the Expressions in the Table

Chord length distribution: Often expressed in the form $f(l)$ over $0 < l < 2R$ . Each model picks chords in a different manner, so the resulting PDFs differ.
$\langle l \rangle$ : 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 $R$ .
Variance( $l$ ): The second central moment minus the square of the first moment, i.e. $\langle l^2\rangle - \langle l\rangle^2$ .

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

If you pick chord midpoints near the sphere’s center more frequently (Method 4) vs. near the surface, you’ll get systematically longer chords on average.
Conversely, if chord endpoints are restricted to the surface (Method 1), you get a different average chord length.

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—these are somewhat standard results from integral geometry or direct integration.

4. Key Takeaway

The table is summarizing known formulas for mean chord length (and chord-length variance) under various “random chord” selection rules in a (presumably) 3D sphere. The references to $\beta_1$ , etc., capture how the expected chord length 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.

Relation to Known Equations

All these cases are variants of the “Bertrand problem” generalized to 3D (sometimes called the Bertrand–Sylvester problem), where chord generation method 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.

References & Context

Solomon (1978) “Geometric Probability” discusses chord problems in 2D and 3D, chord distributions in spheres, and “random secant” approaches.
Bertrand’s paradox (originally for circles) generalizes to spheres, leading to multiple distinct “random chord” definitions.
Integral geometry (e.g., Santalo, Chern, etc.) also addresses chord distributions in higher dimensions.

Final Summary

Your table compares four chord-generation methods in a sphere (labeled #1, #3, #4, #8). Each method yields a different probability distribution of chord lengths, with different mean chord length ( $\beta_1$ ) and variance. Some methods produce the same result (cases #1 and #8 both give $\beta_1 = 4/3$ ), others differ ( $\beta_1 = \pi/2$ , etc.). This underscores how the notion of “random chord” can vary, each interpretation requiring a distinct geometric probability analysis.