Question about the equation of a (two-sheeted) hyperboloid?

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Ineedyou

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OK, so everywhere I look I find that the equation of a (two sheeted) Hyperboloid is:

x2/a2 + y2/b2 - z2/c2 = -1

And of course this can also mean that you can have something like:

(x+5)2 + .... = x2 + 25 + 50x + ....

So you can reduce something of the form A*x2 + B*x + C + D*y2 + E*y + F + G*z2 + H*z + I = J to the equation of a hyperboloid (given that certain criteria are met of course).

What I don't get is the following. I've recently plotted this on Wolfram Alpha:

x^2 + y^2 + 4*z^2 + 16*x*y - 8*x*z + 8*y*z - 104*x +256*y - 142*z = 1

And I found out that this is a two sheeted hyperboloid as well. I don't understand how you can reduce xy or yz or zy to the equation of a hyperboloid. Any help? Thanks in advance
 
I suspect that you'd have to do a rotation of axes in three dimensions. What a mess! :shock:
 
The generalized hyperboloid (both one sheet and two sheet) is given by
(u - u0) A (u - u0)T = 1
where u = (x, y, z), u0 is the center of the hyperboloid [ = (x0, y0, z0) ], A is a 3x3 matrix and the superscript T indicates transpose. See,
http://en.wikipedia.org/wiki/Hyperboloid
for example.

Note that if A is the diagonal matrix with elements 1/a2, 1/b2 , and -1/c2 that the equation
\(\displaystyle (\frac {x-x_0}{a})^2 + (\frac {y-y_0}{b})^2 - (\frac {z-z_0}{c})^2 = 1\)
results. If A is the diagonal matrix with elements -1/a2, -1/b2 , and 1/c2 then the equation
\(\displaystyle (\frac {x-x_0}{a})^2 + (\frac {y-y_0}{b})^2 - (\frac {z-z_0}{c})^2 = -1\)
results.
 
Ishuda said:
The generalized hyperboloid (both one sheet and two sheet) is given by
(u - u0) A (u - u0)T = 1
where u = (x, y, z), u0 is the center of the hyperboloid [ = (x0, y0, z0) ], A is a 3x3 matrix and the superscript T indicates transpose. See,
http://en.wikipedia.org/wiki/Hyperboloid
for example.

Note that if A is the diagonal matrix with elements 1/a2, 1/b2 , and -1/c2 that the equation
\(\displaystyle (\frac {x-x_0}{a})^2 + (\frac {y-y_0}{b})^2 - (\frac {z-z_0}{c})^2 = 1\)
results. If A is the diagonal matrix with elements -1/a2, -1/b2 , and 1/c2 then the equation
\(\displaystyle (\frac {x-x_0}{a})^2 + (\frac {y-y_0}{b})^2 - (\frac {z-z_0}{c})^2 = -1\)
results.
Click to expand...

Yes, I read that too from wikipedia but I don't really understand how I'm supposed to get something concrete out of it. In my previous example for instance, how do I reduce that equation to (u - u0) A (u - u0)T = 1 ?.. how do I have to pick u, u0 and who is A?

How did you arrive at that...isn't it x^2 + 10x + 25?
Click to expand...


Yes, sorry, my typo. I wasn't really paying attention to that since the numbers weren't all that relevant to the idea I was trying to illustrate.
 
Ineedyou said:
Yes, I read that too from wikipedia but I don't really understand how I'm supposed to get something concrete out of it. In my previous example for instance, how do I reduce that equation to (u - u0) A (u - u0)T = 1 ?.. how do I have to pick u, u0 and who is A?



Yes, sorry, my typo. I wasn't really paying attention to that since the numbers weren't all that relevant to the idea I was trying to illustrate.[/COLOR]
Click to expand...
A is a 3x3 matrix and u is a row vector, u = (x y z). The quantity (u - u0) A (u - u0)T multiplied out will give the general equation
a (x-x0)2 + b (x-x0)(y-y0) + c (x-x0)(z-z0) + d (y-y0)2 + e (y-y0)(z-z0) + f (z-z0)2 + g = 0
where a, b, c, d, e, f, and g depend on the elements of A, i.e. a = a11, b = a21 + a12, etc. (assuming I'm remembering correctly). If you expand that equation you will get additional terms involving x, y, and, z for 'an even more general equation'. Note that the generalized hyperboloid can be represented that way does not mean that all equations of that sort are hyperboloids.

Just as a two dimension matrix applied in this manner represents a rotation and stretching/shrinking of a two space equation/object, the 3x3 matrix represents a rotation in three space and a stretching/shrinking of a three space equation. A nice intro, IMO, for 2D can be found at
http://www.willamette.edu/~gorr/classes/GeneralGraphics/Transforms/transforms2d.htm

As far as how to take a general equation like
x^2 + y^2 - 4*z^2 - 104*x +256*y - 136*z = 1
(a more simple example of a one-sheeted hyperboloid) and translate it to the (u - u0) A (u - u0)T is to explicitly write out the equations for the a, b, c, etc. above in terms of the elements of A and back solve. Or just start playing around, i.e.
x2 - 104 x = (x - 52)2 - 522
y2 + 256 y = (y + 128)2 - 1282
4 z2 + 136 z = 4 (z + 17)2 - 4 * 172
so we have
(x - 52)2 + (y+128)2 - 4 (z+17)2 = 1 + ...
 
Ishuda said:
A is a 3x3 matrix and u is a row vector, u = (x y z). The quantity (u - u0) A (u - u0)T multiplied out will give the general equation
a (x-x0)2 + b (x-x0)(y-y0) + c (x-x0)(z-z0) + d (y-y0)2 + e (y-y0)(z-z0) + f (z-z0)2 + g = 0
where a, b, c, d, e, f, and g depend on the elements of A, i.e. a = a11, b = a21 + a12, etc. (assuming I'm remembering correctly). If you expand that equation you will get additional terms involving x, y, and, z for 'an even more general equation'. Note that the generalized hyperboloid can be represented that way does not mean that all equations of that sort are hyperboloids.

Just as a two dimension matrix applied in this manner represents a rotation and stretching/shrinking of a two space equation/object, the 3x3 matrix represents a rotation in three space and a stretching/shrinking of a three space equation. A nice intro, IMO, for 2D can be found at
http://www.willamette.edu/~gorr/classes/GeneralGraphics/Transforms/transforms2d.htm

As far as how to take a general equation like
x^2 + y^2 - 4*z^2 - 104*x +256*y - 136*z = 1
(a more simple example of a one-sheeted hyperboloid) and translate it to the (u - u0) A (u - u0)T is to explicitly write out the equations for the a, b, c, etc. above in terms of the elements of A and back solve. Or just start playing around, i.e.
x2 - 104 x = (x - 52)2 - 522
y2 + 256 y = (y + 128)2 - 1282
4 z2 + 136 z = 4 (z + 17)2 - 4 * 172
so we have
(x - 52)2 + (y+128)2 - 4 (z+17)2 = 1 + ...
Click to expand...

Alright, but.. I see one problem with this. If you expand a (x-x0)2 + b (x-x0)(y-y0) + c (x-x0)(z-z0) + d (y-y0)2 + e (y-y0)(z-z0) + f (z-z0)2 + g = 0, you get ax2 + dy2 + fz2 + bxy + cxz + eyz - x (2ax0 + by0 + cz ) - y (bx0 + 2dy0 + ez0) - z (cx0 + ey0 + 2fz0) + [some constant term defined by a,b,c,d,e,f,x0,y0,z0] (by constant I mean which does not depend on x,y or z). Therefore, every combination of a,b,c,d,e,f,x0,y0,z0 uniquely defines a constant. But that doesn't seem right since (according again to Wolfram alpha), both x^2 + y^2 + 4*z^2 +16*x*y - 8*x*z + 8*y*z - 104*x + 256*y - 142*z - 600 = 0 and x^2 + y^2 + 4*z^2 +16*x*y - 8*x*z + 8*y*z - 104*x + 256*y - 142*z - 1422 = 0 are two-sheeted hyperboloids and they ony differ through their constant. So I'm assuming the generalized equation isn't a (x-x0)2 + b (x-x0)(y-y0) + c (x-x0)(z-z0) + d (y-y0)2 + e (y-y0)(z-z0) + f (z-z0)2 + g = 0, but rather a (x-x0)2 + b (x-x0)(y-y0) + c (x-x0)(z-z0) + d (y-y0)2 + e (y-y0)(z-z0) + f (z-z0)2 + g = [some constant] ?

Another issue I have is how do you determine matrix A given the equation of a hyperboloid? From what I can see you can only determine a11,a22 and a33 but other than that you can only determine a12+a21 (and other similar terms), not a12 itself. From what I've read on wikipedia, to prove that a certain equation is indeed the equation of a two-sheeted hyperboloid you would have to prove that it satisfies (u - u0) A (u - u0)T and that A has one positive eigenvalue and two negative eigenvalues, but I don't really know how I can do that without having matrix A properly determined.

I really appreciate all of the help you have provided thus far by the way :)
 
Lets continue with that example I started
so we have
(x - 52)2 + (y+128)2 - 4 (z+17)2 = 1 + ... = some constant k
If k is not zero we can divide through by |k| and get
\(\displaystyle \frac{(x-52)^2}{|k|} + \frac{(y+128)^2}{|k|} - \frac{(z+17)^2}{|k|/4} = \pm1\)
depending on whether k is positive or negative. Thus the constant term in your general equation helps determine the a, b, and c for the quadratic surface and not (always) the type of quadratic surface. Note here in this example the a, b, and c are, respectively \(\displaystyle \sqrt{|k|+},\space \sqrt{|k|},\text{ and}\space \sqrt{|k|}/2\) and the diagonal elements of the matrix A are 1/a2, 1/b2, and -1/c2.

Note that I started out without a general cross product term, i.e. neither an x y, x z, nor z y term. If you don't have those terms, then 'the completion of squares' technique works well for determining the equation of the (u-u0) A (u-u0)T type. If you have those terms, then by a suitable rotation of co-ordinates they can be eliminated and that generally process can be expressed as, well as staple put it, "What a mess!". However, a walk through with the formulas, can be found at
http://mathworld.wolfram.com/QuadraticCurve.html
First eliminate the x y cross term (as outlined in the link), then eliminate the x z cross term, then the y z cross term. Finally, go through the 'completion of squares' to get to the diagonal matrix is some rotated co-ordinate system to determine the type of surface.

Although the standard form for a hyperboloid is
\(\displaystyle \frac{x^2}{a^2} + \frac{y^2}{|k|} - \frac{z^2}{c^2} = \pm1\)
the minus sign could go on any of the other variables (swap the plus and minus sign on x and z or on y and z). This swapping would just change the axis of the hyperboloid.
 
Oh so I can't simply expand (u-u0) A (u-u0)T and equalize the coefficients?.. I need to do a rotation of axis in three dimensions?.. well, this really is a mess :(.. but thanks again for the help :)
 
Ineedyou said:
Oh so I can't simply expand (u-u0) A (u-u0)T and equalize the coefficients?.. I need to do a rotation of axis in three dimensions?.. well, this really is a mess :(.. but thanks again for the help :)
Click to expand...
The two are equivalent but one may be easier than the other depending on circumstances.
 
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