On Jun 22, 10:33*am, Tim Wescott <t...@seemywebsite.com> wrote:
> On Mon, 22 Jun 2009 01:43:28 -0700, illywhacker wrote:
>
> On Jun 20, 6:27*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. *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. *It is a two
> > dimensional figure enclosed by the locus of all points that are a
> > distance of r away from the center point. *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). *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.
> *It
> > has a length of 2 r and a -- uh -- oh never mind.
>
> > Now hopefully I have some momentum: *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. *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? *Wrong? *Anyone done this calculation before?
>
> > Thanks. *(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- Hide quoted text -
>
> - Show quoted text -

Tim,
I have no idea if this will help, but you might try looking at a
couple of things on quaternion calculus.

http://www.geometrictools.com/Docume...uaternions.pdf

http://www.euclideanspace.com/math/d...ulus/index.htm

These might get you to the point where you can read the book ;>)

Maurice Givens