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Cyphoid

n-dimensional ellipse

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Hi there, I was hoping to pick a mathematician's brain: Prof wants us to derive the equation for (initially) a 4-dimensional ellipse. I guessed that, rather than creating a volume of rotation with a disk (2 dimensional) around the 3rd dimension axis (like I did to derive the volume), I would rotate the Volume around the 4th dimension (w) axis. I came up with 16/9 * pi*(abcd), where a, b, c, d are magnitude values in their respective axis. I expected the pi*abcd, which follows logically after ellipsoid volume (4/3*pi*abc) and area (pi * ab), but I wanted to know if the coefficient was correct. I would guess that it is correct, being a factor of 4/3, but I would feel much better with a second opinion. If I'm right, it should make writing the general form for an n-dimensional ellipsoid very easy, ha. Thanks in advance

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I have no idea what you're talking about, but isn't the standard form n-ellipse just going to be the same with more axis. Something like (X1-C1)^2*A1+(X2-C2)^2*A2+...(Xn-Cn)^2*An = d

Also, homework questions don't go over too well here.

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I think the OP is asking about the formula for the hyper-volume of a hyper-ellipsoid (I guess that's what it would be called) in 4 dimensions that has axes of size a, b, c and d. Of course, the formula will be "hyper-volume of a hyper-sphere of radius 1 in 4 dimensions" * abcd, because an affine transformation with determinant abcd maps the unit hyper-sphere to the hyper-ellipsoid.

The magic number in dimension 4 is pi^2/4.



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Thanks for the tip. I ran the numbers again with a buddy (during physics, heh), and you are right, although it appears that it ought to be pi^2 / 2, rather than pi^2 / 4. Whatever the case, I will straighten it out and maybe check some more resources.

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Quote:
Original post by Cyphoid
Thanks for the tip. I ran the numbers again with a buddy (during physics, heh), and you are right, although it appears that it ought to be pi^2 / 2, rather than pi^2 / 4. Whatever the case, I will straighten it out and maybe check some more resources.


Oh, you are right. I was quoting that number from memory, but my memory was wrong.

The general formula is pi^(n/2)/(n/2)!, where n is the dimension. For odd dimensions, use the gamma function to define the factorial.

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