The VERY, VERY general equation of an ellipsoid - Who knows it?

[Umbra Lupis]

I have been asking/looking around for the general equation of an ellipsoid and I am unable to find it anywhere.

Does anyone know what it is?

BTW: What I mean by the general equation of an ellipsoid, one that can be rotated in any way, that is 2 angles of rotation and one that does not have to be centered at the origin. - I know the one for a general ellipse moved from the center. So I don't want this!
$$\frac{(x-x_c)^2}{a^2} + \frac{(y-y_c)^2}{b^2} + \frac{(z-z_c)^2}{c^2} = 1$$

If possible could it be in Implicit Cartesian or Spherical Polars form?

Thanks for any help

 

[g_edgar]

A polynomial of degree 2 in variables x,y,z is ellipsoid or hyperboloid or paraboloid.

$$ax^2+by^2+cz^2+dxy+eyz+fzx+gx+hy+iz+j=0$$

There are some inequalities on the coefficients that determine which of the three types it is.

 

[Umbra Lupis]

Wow dude thanks for the help!

However, I was wondering if you would not happen to know the inequalities for an ellipsoid? Also, if possible you would not happen to have the equations relating your constants:

a, b, c, d, f, g, h, i, j to the:

x_c: center of ellipse in the x-direction
y_c: center of ellipse in the y-direction
z_c: center of ellipse in the z-direction

x_r: equatorial radius in the x-direction
y_r: equatorial radius in the y-direction
z_r: polar radius in the z-direction

$$\gamma$$: the angle of rotation in the xy plane (starting from the positive x-axis, where $$0 \leq \gamma < 2\pi$$)
$$\eta$$: the angle of rotation from the positive z-axis ($$0 \leq \eta \leq \pi$$)

Though these last two could be expressed as the components of vectors in the x, y, z direction.

At any rate thanks a lot for the equation! You have helped a lot! The stuff above is not such a big deal: though if you happened to have it on hand it would really useful!

 

[Ben Niehoff]

Do you know how to use rotation matrices? The equation of an ellipse can be written as a matrix equation:

$$(\mathbf{x - c})^T \mathbf{A} (\mathbf{x - c}) = 1$$

where x is a column vector (x, y, z), c is a column vector representing the center of the ellipse (xc, yc, zc), and A is a square, symmetric matrix. In your equation in the OP, your matrix A is diagonal, with entries (1/a^2, 1/b^2, 1/c^2).

Now, to rotate your equation to an ellipse in a general orientation, you just need to apply a rotation matrix R as follows:

$$(\mathbf{x - c})^T \mathbf{R}^T \mathbf{A} \mathbf{R} (\mathbf{x - c}) = 1$$

For an ellipse, the matrix A is always positive definite. For a hyperboloid, it may have signature (1, -1, -1) or (-1, 1, 1).

 

[Umbra Lupis]

Thanks!
The matrix form would actually be really cool to use; but I am ignorant of using matrix geometry. If someone could provide explanations and examples of the following, I might be able to understand what to do:

$$(\mathbf{x - c})^T \mathbf{R}^T \mathbf{A} \mathbf{R} (\mathbf{x - c}) = 1$$

What does the $$^T$$ mean in the $$(\mathbf{x - c})^T$$ - inverted matrix?
What does a rotation matrix,$$\mathbf{R}$$, look like - say I wanted to rotate it $$\theta$$ degrees in the xy-plane and $$\phi$$ from the z-axis, what would it look like?
Finally what is a square, symmetric matrix, $$\mathbf{A}$$, what does it do?

Thanks once again to anyone who helps out!

 

[Ben Niehoff]

T means Transpose. For rotation matrices and symmetric matrices, try typing those phrases into Google.

 

[Umbra Lupis]

Thanks dude!