A General Approach to Chord Length Distributions Applied to a Hemisphere

Abstract

General formulas for chord and related joint density distributions of convex volumes are derived and applied to the hemisphere. The resulting single integral for the joint chord length and beam angle distribution is evaluated analytically. The chord length distribution is obtained by the same methods by evaluating a double integral. Both results are presented graphically, and accurate agreement with norm and average chord length is exhibited. The chord distribution agrees with previous work. While not new, the general approach used here appears to be of more extensive applicability than current methods, potentially applying to all chord and related distributions of all convex volumes.