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From: Cheng Cosine on 11 Apr 2010 08:44
Hi:

We have in 2D polar coordinates and in 3D spherical coordinates. But
do we have other parametric expressions for a hyper-ball in a
dimension higher than 3?

Thanks,
From: Mike Terry on 11 Apr 2010 10:41
"Cheng Cosine" <asecant(a)gmail.com> wrote in message
news:090ebac8-92fd-4dc2-905d-3e56a9960190(a)y14g2000yqm.googlegroups.com...
> Hi:
>
> We have in 2D polar coordinates and in 3D spherical coordinates. But
> do we have other parametric expressions for a hyper-ball in a
> dimension higher than 3?
>
> Thanks,

You can think of the 3D spherical coordinate scheme as being built on top of
the 2D scheme: to get the angles for a vector V in 3D, project the vector
onto a 2D plane to get vector V_p. The direction for V_p within the plane
is expressed by a single angle (the 2D polar coordinate angle), and then we
add the angle between V and V_p to get the second angle used in spherical
coordinates.

This approach extends in an obvious fashion to higher dimensions. Each
dimension will add another angle coordinate, which is the angle between the
desired vector and its projection onto the hyper-plane of one lower
dimension.

Regards,
Mike.


From: Igor on 11 Apr 2010 10:51
On Apr 11, 8:44 am, Cheng Cosine <asec...(a)gmail.com> wrote:
> Hi:
>
>  We have in 2D polar coordinates and in 3D spherical coordinates. But
> do we have other parametric expressions for a hyper-ball in a
> dimension higher than 3?
>
>  Thanks,

Here it is for the generalized hypersphere:

http://en.wikipedia.org/wiki/Hypersphere

From: Chip Eastham on 11 Apr 2010 11:15
On Apr 11, 8:44 am, Cheng Cosine <asec...(a)gmail.com> wrote:
> Hi:
>
>  We have in 2D polar coordinates and in 3D spherical coordinates. But
> do we have other parametric expressions for a hyper-ball in a
> dimension higher than 3?
>
>  Thanks,

Hi, Cheng:

One name for this is "direction cosines"; Google for it and
consider "direction" as a middle name.

Given a point X = (x_1,...,x_N) in (real) N-dimensional space,
write X = ||X|| * (c_1,...,c_N), and c_i = cos(theta_i) while
||X|| plays the role of radius (from the origin). The cosines
satisfy:

SUM c_i^2 = 1

so there's actually one less degree of freedom than might at
first appear. theta_i can be interpreted as the angle between
a ray from the origin through X and the positive-directed ith
coordinate axis.

regards, chip
From: spudnik on 12 Apr 2010 15:38
for spherical coordination,
I am only able to do it in tripolars.

> One name for this is "direction cosines"; Google for it and
> consider "direction" as a middle name.

thus:
oh, yes; les God-am points de Brocard --
what were they, and are you using trilinears, or
is it "intrinsic?"

> if sin^3(x) = sin(A-x)sin(B-x)sin(C-x) and A+B+C=pi,
> then cot(x) = cot(A) + cot(B) + cot(C).

--Light: A History!
http://21stcenturysciencetech.com
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