On Mon, 22 Jun 2009 01:43:28 -0700, illywhacker wrote:

On Jun 20, 6:27Â*am, Tim Wescott <t...@seemywebsite.com> wrote:
> A sphere is a three dimensional figure enclosed by the locus of all
> points that are of distance r (called the radius) away from the center
> point. Â*It has a volume of 4/3 pi r^3 and a surface area of 4 pi r^2.
>
> A circle is the two-dimensional analog of a sphere. Â*It is a two
> dimensional figure enclosed by the locus of all points that are a
> distance of r away from the center point. Â*It has an area of pi r^2 and
a
> circumference of 2 pi r.
>
> A line is a one-dimensional analog of a sphere (OK, I'm reaching, but
> bear with me). Â*It is a one dimensional figure enclosed by the locus of
> all (two) points that are a distance of r away from the center point.
Â*It
> has a length of 2 r and a -- uh -- oh never mind.
>
> Now hopefully I have some momentum: Â*The four dimensional analog of a
> sphere (commonly called a hypersphere) is a four dimensional figure
> enclosed by the locus of all points that are a distance of r away from
> the center point. Â*It contains an amount of space (not a volume,
> certainly -- hypervolume?) equal to 1/2 pi^2 r^4, and has a 'surface'
> volume of 2 pi^2 r^3.
>
> I've done the math in a way that seems obvious to me (integrate the
> volume of the sphere that forms the surface of the hypersphere as one of
> the dimensions varies from -r to r), and that pi^2 just _belongs_ there,
> no matter how much it surprises me.
>
> So, am I right? Â*Wrong? Â*Anyone done this calculation before?
>
> Thanks. Â*(and no, this isn't idle speculation; I actually need to know
to
> solve a DSP problem I'm wrestling with).
:
:'Directional statistics'.
:
:illywhacker;

A short web search shows that this is a book length subject, and one that
may require more than one book to get ready to read the book on the
subject.

I'll probably have to muddle through as best as I can with this (I'm
starting to think that some sort of uniform PDF on a 3-D plane, mapped to
the hypersphere surface, may get me sufficient performance without making
my brain explode).

So -- got any book recommendations? I've got statistics, estimation and
detection theory, a bit of functional analysis, and Kalman filtering
under my belt, but it's mostly from an engineering perspective rather
than a 'math wonk' perspective. It would be nice if I could get just one
book that does a good job of teaching the run-in (what the hell is a
Riemann Manifold?), rather than having to go buy books just so I can read
the book.

--
www.wescottdesign.com