hyperconic sections

Higher-dimensional geometry (previously "Polyshapes").

hyperconic sections

Postby elpenmaster » Mon Sep 20, 2004 5:00 am

What are the conic sections for 3d cartesian plane? for a 3d plane, they are all the cross sections of a plane and a double cone (hourglass shape), so for 3d i assume the would be intersections of realm and tetrahourglass shape :P
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Postby pat » Mon Sep 20, 2004 4:34 pm

The hyper(double)cone would be:
x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = ±w

An arbitrary realm would be:
ax + by + cz + e = w

The intersection, then, would be all points satisfying:
x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = ±( ax + by + cz + e )

In the particular case where the realm is perpendicular to the w axis, the result is:
x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = |e|
This is a 3d-sphere (S<sup>2</sup>). This is just as one would expect given that the (3d-)conic section for a plane perpendicular to the z-axis is a circle.

For other planes, I'm not having an easy time making anything meaningful out of the equations just yet.
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Postby pat » Mon Sep 20, 2004 7:37 pm

Another way of formulating the 2-D conic sections is that they are all points such that the ratio of the distance from a fixed point F (the focus) and the distance from a fixed line D (the directrix) (not through F) is constant.

Of course, moving this all up a dimension, we are looking for all points such that the ratio of the distance from a fixed point F and the distance from a fixed realm D (not containing F) is constant.

WLOG, assume that D is the realm w = 0 and that F is the point ( 0, 0, 0, p ) and the said ratio is e > 0. Then, the general conic section is all points ( x, y, z, w ) satisfying:
e<sup>2</sup>w<sup>2</sup> = x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + ( w - p )<sup>2</sup>
Image
e = √2/2, p = 1

In the special case where e = 1, this is the parabolic shape:
w = ( x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + p<sup>2</sup> ) / p
Image
e = 1, p = 1

In the case where 0 < e < 1, this is a non-spherical ellipsoid. In the case where e > 1, this is a hyperbolic solid.
Image
e = √6/2, p = 1

Defined this way, the conic sections do not get any more interesting as one goes up in dimensions. There are only two free parameters, the focal point's distance from the plane and the ratio. Now, things would definitely get more interesting if we let the directrix D be any flat space of dimension less than n and we let F be any flat space of dimension less than that of D. These wouldn't be conic any longer, but they'd get more interesting.
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Postby elpenmaster » Mon Sep 27, 2004 6:14 am

I'm not great with equations, but i was looking for a more simple aanswer, like how many conic sections are there in 3d? what would they be? is there a formula for how many there are i an X-d cartesian thingy? :)
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Postby pat » Mon Sep 27, 2004 2:42 pm

As near as I can tell, if your cone has cross-sections (perpendicular to the height) which are spherical (n-1-dimensional spheres), then there are the same number of conic sections with the same basic shapes no matter how many dimensions you have (as long as there are at least two dimensions).

And, as near as I can tell, the answer is the same if you define the sections in terms of a focus and a directrix.
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