# N-sphere chord length distribution

arXiv: https://arxiv.org/abs/1411.5639 · HTML: https://mlsanity.com/paper/1411.5639/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Nov 2014.

## Built on by (most cited first, 1 of 1)

- [Depth Separation in Norm-Bounded Infinite-Width Neural Networks](https://mlsanity.com/paper/2402.08808/) (2024)

## Abstract (arXiv)

This work studies the chord length distribution, in the case where both ends lie on a N-dimensional hypersphere (N ≥ 2). Actually, after connecting this distribution to the recently estimated surface of a hyperspherical cap citeSLi11, 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.