Area of an EllipsoidDate: 09/28/2001 at 11:32:39 From: Jason Barnabas Subject: Area of an ellipsoid How do you calculate (or even estimate) the area of an ellipsoid that is neither oblate nor prolate (a<>b<>c<>a)? Date: 09/28/2001 at 16:34:09 From: Doctor Mitteldorf Subject: Re: Area of an ellipsoid The solution when the two axes are equal can be found in the Dr. Math FAQ: Ellipsoid, Torus, Spherical Polygon Formulas http://mathforum.org/dr.math/faq/formulas/faq.ellipsoid.html I'm more comfortable with numerical methods, and that's how I can tell you to solve it. I'll just outline the procedure - if you need more details, please write back. Let your ellipsoid be (x/a)^2 + (y/b)^2 + (z/c)^2 = 1. Set up a double integral over the portion of the x-y plane enclosed by the z = 0 projection of the ellipsoid: X integral from -a to + a Y integral from -b(1-(x/a)^2) to b(1-(x/a)^2) The integrand is the ratio of the size of a little bit of area in the ellipsoid to dxdy in the xy plane; this ratio is 1/cos(phi), where phi is the angle between the local normal to the ellipsoid and the z axis. To find the normal vector, use the fact that the function F = (x/a)^2 + (y/b)^2 + (z/c)^2 is constant all along the surface. Therefore, the direction in which F changes most steeply (del F) is the local normal to the surface. The components of del F are, respectively, the partial derivatives of F with respect to x, y and z. Once you have this vector, you need to divide by its magnitude to make a unit vector, and then the z component is the very cosine that you are seeking. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ Date: 09/28/2001 at 16:38:47 From: Doctor Rob Subject: Re: Area of an ellipsoid Thanks for writing to Ask Dr. Math, Jason. The exact area cannot be found in closed form in terms of familiar functions of a, b, and c, and familiar constants. It can be computed in terms of the value of the integral of an elliptic integral (!). I know of no good way to approximate the surface area in symbolic form. Once fixed values are given to a, b, and c, one may use a convergent power series. One may also do numerical integration to evaluate the elliptic integral, and then its integral. In my view, it is a minor miracle that the prolate and oblate spheroids have closed-form formulas, given the above facts. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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