[Paper Review] On Recursive Random Prolate Hyperspheroids

Jonathan D. Gammell, Siddhartha S Srinivasa+1 more|arXiv (Cornell University)|Mar 29, 2014

Mathematics and Applications3 references3 citations

TL;DR

This paper investigates a recursive process generating random prolate hyperspheroids with fixed foci, where each new hyperspheroid passes through a uniformly random point sampled from the volume of the previous one. It derives the expected transverse diameter of the next hyperspheroid and proves almost-sure convergence to the minimum diameter (distance between foci), with a linear convergence rate of (n−1)/(n+1) in n-dimensional space.

ABSTRACT

This technical note analyzes the properties of a random sequence of prolate hyperspheroids with common foci. Each prolate hyperspheroid in the sequence is defined by a sample drawn randomly from the previous volume such that the sample lies on the new surface (Fig. 1). Section 1 defines the prolate hyperspheroid coordinate system and the resulting differential volume, Section 2 calculates the expected value of the new transverse diameter given a uniform distribution over the existing prolate hyperspheroid, and Section 3 calculates the convergence rate of this sequence. For clarity, the differential volume and some of the identities used in the integration are verified in Appendix A through a calculation of the volume of a general prolate hyperspheroid.

Motivation & Objective

• To model a stochastic recursive process where each prolate hyperspheroid is defined by a random point sampled from the volume of the prior one.
• To analyze the expected value of the transverse diameter of the new hyperspheroid given uniform sampling over the previous one’s volume.
• To determine the convergence behavior and rate of the sequence of transverse diameters toward the minimum diameter (distance between foci).
• To rigorously verify the differential volume element and hyperspheroidal coordinate system using integration identities and beta functions.
• To establish the theoretical foundation for recursive geometric sampling in high-dimensional spaces with applications in robotics and estimation.

Proposed method

• Defines a prolate hyperspheroid coordinate system in R^n using parameters μ, ν, and ψ₁,…,ψₙ₋₂, with foci at (±a, 0, ..., 0).
• Derives the differential volume element dV using scale factors from the Jacobian of the coordinate transformation, resulting in dV = h_μ h_ν dμ dν ∏(h_ψᵢ dψᵢ).
• Computes the expected transverse diameter E[dᵢ₊₁] by integrating over the uniform distribution within the previous hyperspheroid, using the identity d = d_min cosh μ.
• Applies known indefinite integrals of hyperbolic functions (e.g., ∫cosh x sinhⁿx dx = sinhⁿ⁺¹x / (n+1)) to evaluate the expectation.
• Uses the beta function and properties of the unit n-ball volume to verify the volume formula for a general prolate hyperspheroid.
• Derives the convergence rate η by differentiating E[dᵢ₊₁] with respect to dᵢ and evaluating at dᵢ = d_min, yielding η = (n−1)/(n+1).

Experimental results

Research questions

• RQ1 What is the expected transverse diameter of a prolate hyperspheroid generated by sampling a point uniformly from the volume of a previous hyperspheroid with the same foci?
• RQ2 How does the sequence of transverse diameters behave over repeated iterations—does it converge, and to what value?
• RQ3 What is the rate of convergence of the transverse diameter sequence toward the minimum diameter d_min?
• RQ4 How can the differential volume element in prolate hyperspheroidal coordinates be rigorously derived and verified?
• RQ5 Can the volume of a general prolate hyperspheroid be recovered via integration in this coordinate system?

Key findings

• The expected transverse diameter of the next hyperspheroid is E[dᵢ₊₁] = (n dᵢ² + d_min²) / ((n+1) dᵢ), derived from uniform sampling over the volume of the previous one.
• The transverse diameter sequence almost surely converges to d_min, the distance between the fixed foci, due to the zero-probability of sampling exactly on the surface.
• The convergence rate is linear and given by η = (n−1)/(n+1), which is strictly between 0 and 1 for all n ≥ 2.
• The derived volume formula for a prolate hyperspheroid, V = ζₙ d (d² − d_min²)^(n−1)/² / 2ⁿ, is rigorously verified using beta functions and hyperbolic integral identities.
• The differential volume element is confirmed via coordinate transformation and scale factor computation, with consistent results across multiple integration paths.
• The analysis confirms that the recursive sampling process leads to a monotonic decrease in diameter almost surely, with predictable asymptotic behavior in high dimensions.