Cheng Cosine <asecant@gmail.com> wrote in message <761f880b8c7b48e58f407f9f12647988@x3g2000yqd.googlegroups.com>...
> Thanks Roger; it becomes clearer to me now. :)
>
> My original thought is to have some visualizing tool to understand
> the pball, and 2D is the most simple one to start with. But,
> according to your 1st replying, it seems that we do not have
> parametric formulations for pball in a dimension higher than 3? Thus,
> we do not even have a way to generate numerical data of pball in
> higher spatial dimensions, and then strive to figure out a way to
> visualize the data?
>
> In 3D we have spherical parametric expression, but I do not recall
> one for 4D or higher D's.
>
> Thank you,

I think it is time to make some remarks about terminology here. Mathematicians, at least the ones I am familiar with, distinguish between the terms 'circle' or 'sphere' on the one hand and 'disk' and 'ball' on the other hand. A circle in two dimensions and a sphere in three dimensions consist of all points whose distance from a central point is equal to a given constant, the radius. A disk in two dimensions or a ball in three dimensions consist of all points whose distance from a central point is less than or equal to its radius. In other words a (closed) ball would be all points on, or within the interior of, a sphere. A disk is all points on, or within the interior of, a circle. A sphere would be the surface of a ball. A circle would be the periphery of a disk.
For this reason it would seem more appropriate with respect to a pnorm metric to call the objects you have been dealing with pcircles, pspheres, or phyperspheres, and you could call the points on or within their interiors pdisks or pballs. In that terminology the point sets I generated yesterday would have to be called pcircles rather than pdisks or pballs.
In answer to your question above, in four dimensions one can define a system of hyperspherical coordinates, t1, t2, t3, and r, which are related to the cartesian coordinates x1, x2, x3, and x4 by the equations
x1 = r*sin(t3)*sin(t2)*sin(t1)
x2 = r*sin(t3)*sin(t2)*cos(t1)
x3 = r*sin(t3)*cos(t2)
x4 = r*cos(t3)
where 0 <= t1 <= 2*pi, 0 <= t2 <= pi, 0 <= t3 <= pi, and 0 <= r. The inverse of these is
t1 = atan2(x1,x2)
t2 = atan2(sqrt(x1^2+x2^2),x3)
t3 = atan2(sqrt(x1^2+x2^2+x3^2),x4)
r = sqrt(x1^2+x2^2+x3^2+x4^2)
which are well defined except in the cases x1 = x2 = 0.
Inspired by this, we can define a four dimensional phypersphere of fixed radius r and centered at the origin using parameters t1, t2, and t3 as follows:
[t1,t2,t3] = ndgrid(0:pi/n:2*pi,0:pi/n:pi,0:pi/n:pi);
x1 = r*sign(sin(t1)).*abs(sin(t3).*sin(t2).*sin(t1)).^(2/p);
x2 = r*sign(cos(t1)).*abs(sin(t3).*sin(t2).*cos(t1)).^(2/p);
x3 = r*sign(cos(t2)).*abs(sin(t3).*cos(t2)).^(2/p);
x4 = r*sign(cos(t3)).*abs(cos(t3)).^(2/p);
Each of the points (x1,x2,x3,x4) thus defined belongs to the indicated phypersphere as can be tested by checking that
(x1.^p+x2.^p+x3.^p+x4.^p).^(1/p) = r
(to within round off accuracy of course.) The big question is, how are you going to display this three parameter mesh in four dimensions? All we can effectively envision are 3D "slices" of this monster.
The generalization of this example to higher dimensions is easily made, but these continue to have the same difficulty that such phyperspheres are very difficult for us creatures to have displayed for us with our limited threedimensional brains.
Roger Stafford
