### So Close, Yet So Far Away

I humbly entreat any analytical intellects of greater constitution than my own (of which, to be sure, there is no dearth) to enlighten me in matters mathematical.

First of all, if I have a 5-dimensional space which consists of a 3-dimensional Euclidean space plus direction (defined by inclination and azimuth) — that is, the set of vectors $(x, y, z, \rho, \eta)$ where $x, y, z \in \mathbb{R}$, $\rho \in [0, \pi]$, and $\eta \in [0, 2\pi)$ — what is that called? Of course, this generalizes to a (2p – 1)-dimensional space with a p-dimensional Euclidean spatial component. I have been calling fuzzy subsets of this space spatial-directional fuzzy sets, but spatial-directional space sounds patently ridiculous.

Second, is it possible to define a useful distance metric in such a space? Chaudhuri and Rosenfeld generalize the Hausdorff distance to arbitrary fuzzy subsets of a metric space, but this is of little use to me if my universal space is not metric. The natural way to define distance between two directions is to use the angle between the corresponding vectors, or similarly a norm on the surface of a torus. The obvious problem is that the numbers used for space and angle bear no relation to one another, so it seems nonsensical to combine them in a single metric (scaling and other such hackery need not apply). Yet, configuration spaces with similar discord among the units of their dimensions abound in engineering. Surely someone has tried to do something like this before?

Jun 15th, 2010
Tags: ,