N-sphere chord length distribution
This work provides theoretical results for geometric distributions in high-dimensional spaces, which is incremental as it builds on prior estimates of hyperspherical cap surfaces.
The paper tackles the problem of deriving the chord length distribution for points on an N-dimensional hypersphere, resulting in closed-form expressions for the probability density and cumulative distribution functions, with dependencies on the dimension N.
This work studies the chord length distribution, in the case where both ends lie on a $N$-dimensional hypersphere ($N \geq 2$). Actually, after connecting this distribution to the recently estimated surface of a hyperspherical cap \cite{SLi11}, closed-form expressions of both the probability density function and the cumulative distribution function are straightforwardly extracted, which are followed by a discussion on its basic properties, among which its dependence from the hypersphere dimension. Additionally, the distribution of the dot product of unitary vectors is estimated, a problem that is related to the chord length.