Mean Chord Length Sphere

Series representations of the chord length and free path distribution functions for systems of spheres with a polydispersivity in size in arbitrary dimension d. For starters it should be obvious that the average chord length will scale with r not r 2.

Equations 2 4 And 2 5 Are Two Definitions Of T

mean chord length sphere

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The relation is known in literature weinberg am wigner ep 1958.

Mean chord length sphere. The most probable value of l is at the right end point ie mode of the l distribution 2a. The reason for this discrepancy is different weighting. A chord of a circle is a straight line segment whose endpoints both lie on the circle.

A secant line or just secant is the infinite line extension of a chord. The mean value of l is computed rather simply as. The central angle between the two points can be determined from the chord length.

I did it with a double integral over the surface of the sphere and got the correct answer. The probability of measuring a value of l greater than the mean is given by. The relation is known in literature weinberg am wigner ep 1958.

For the special case of spatially uncorrelated spheres ie fully penetrable spheres we evaluate exactly the. I suspect the problem with your method has something to do with the way a bunch of circumferences add up to become the whole surface of the sphere. For example since by default normalised vectors lie on a hypersphere of radius r 1 the n sphere chord length distribution could be used as a randomness benchmark to give a hint whether a set of normalised vectors is selected uniformly and independently.

From it follows that the average chord length for a sphere with radius a is 4a3. More generally a chord is a line segment joining two points on any curve for instance an ellipse. The great circle distance is proportional to the central angle.

However when we calculated the average chord length for a sphere assuming an isotropic flux distribution the result was equal to the radius a. By elaborating on the meaning of the average chord length in reactor physics it is shown that the average chord length for a convex body in an isotropic flux is given by the bodys volume divided by its average projection area. A line through three dimensional space between points of interest on a spherical earth is the chord of the great circle between the points.

Actually after connecting this distribution to the recently estimated surface of a hyperspherical cap. A chord that passes through a circles center point is the circles diameter. N sphere chord length distribution panagiotis sidiropoulos abstract this work studies the chord length distribution in the case where both ends lie on a n dimensional hypersphere n 2.

Finally the most important and intuition defying result of the sphere game is now readily apparent. By elaborating on the meaning of the average chord length in reactor physics it is shown that the average chord length for a convex body in an isotropic flux is given by the bodys volume divided by its average projection area.

The Sphere Game