# Re: Oblate ellipsoid

From: Alberto Fasso' <fasso_at_SLAC.Stanford.EDU>
Date: Fri, 29 Apr 2011 06:19:46 -0700 (PDT)

Yes, it is possible. An ellipsoid is a quadric, so you can use the QUA body.

The equation of a generic ellipsoid with semiaxes a, b, c parallel to the
axes of the coordinates is:
(x-x0)**2/a**2 + (y-y0)**2/b**2 + (z-z0)**2/b**2 -1 = 0

The QUA (see the manual) is defined by 10 numbers:

Axx, Ayy, Azz, Axy, A_xz, Ayz, Ax, Ay, Az, A0

corresponding to the equation:
Axx*x**2 + Ayy*y**2 + Azz*z**2 + Axy*xy + Axz*xz + Ayz*yz +
+ Ax*x + Ay*y + Az*x + A0 = 0

Re-arranging the ellipsoid equation above, the 10 numbers become:
Axx = (b*c)**2
Ayy = (a*c)**2
Azz = (a*b)**2
Axy = Axz = Ayz = 0
Ax = -2*x0*(b*c)**2
Ay = -2*y0*(a*c)**2
Az = -2*z0*(a*b)**2
A0 = (b*c*x0)**2 + (a*c*y0)**2 + (a*b*z0)**2 - (a*b*c)**2

If the ellipsoid is oblate (ellipsoid of rotation around the minor axis),
two of its semiaxes are equal, and larger than the third:
b = c> a (minor axis parallel to x)
or
a = c> b (minor axis parallel to y)
or
a = b> c (minor axis parallel to z)

Finally if your oblate ellipsoid is a generic one with semiaxes not
parallel to the coordinate axes, you could in principle work out
its equation (in which Axy, Axz and Ayz are not equal to zero): but
it is much easier to use one of the above simpler equations, and to rotate
the body by means of a transformation directive (see Manual 8.4.3})

Alberto

On Thu, 28 Apr 2011, adwasil_at_poczta.onet.pl wrote:

> Is there any possibilities to define oblate ellipsoid?
> Best regards