RE: Re: Oblate ellipsoid

Adam,

you are right: the two formulae are equivalent. You have simply divided
all my coefficients by (a*b*c)**2. (There is just a typing error in your Az
coefficient, see below).
I had tried to eliminate all fractions, but I admit that your formula looks
simpler.

I take the liberty to post your mail on the discussion list, together
with this reply of mine. Other users might be interested.

Kind regards,

Alberto

On Wed, 4 May 2011, adwasil_at_poczta.onet.pl wrote:

> Thanks a lot Alberto.
> Your answer was very helpful. I tried to count himself a little bit and I came
> out slightly different formula.
> Axx = 1 / a ** 2
> Ayy = 1 / b ** 2
> Azz = 1 / c ** 2
> Axy = AXZ = Ayz = 0
> Ax =- 2x0 / a ** 2
> Ay =- 2y0 / b ** 2
> Az =- 2y0 / c ** 2<-- obviously this was a typo: not -2y0 but -2z0
> A0 = x0 ** 2 / a ** 2 + y0 ** 2 / b ** 2 + z0 ** 2 / c ** 2 - 1
> They seem to me to be simpler, although probably we describe the same thing. I
> checked them with concrete numbers in the flair code and it seems to work.
> Best regards
> Adam Wasilewski
>
>
> W dniu 2011-04-29 15:19:54 u?ytkownik Alberto Fasso'<fasso_at_slac.stanford.edu> napisa?:
>> 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
>>> Adam Wasilewski
>>
>
>
>
>

-- 
Alberto Fasso`
SLAC-RP, MS 48, 2575 Sand Hill Road, Menlo Park CA 94025
Phone: (1 650) 926 4762   Fax: (1 650) 926 3569
fasso_at_slac.stanford.edu