On Recursive Random Prolate Hyperspheroids

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. 1 Prolate Hyperspheroid Coordinate System Let (x1, x2,..., xn) be the Cartesian coordinates of an Rn coordinate system, then we can define a prolate hyperspheroid coordinate system, (µ, ν, ψ1, ψ2,..., ψn−2), parameterized on a as x1 =: a coshµ cos ν, (1) x2 =: a sinhµ sin ν cosψ1, x3 =: a sinhµ sin ν sinψ1 cosψ2, xn−1 =: a sinhµ sin ν sinψ1 sinψ2... sinψn−3 cosψn−2