Volume of Ellipsoidal Cap
Date: 04/11/2001 at 11:19:54 From: R Shanks Subject: prob309885_02molume of ellispoidal cap I am doing research on cancer and need a way to properly determine the volume of tumors in lab animals. Such a tumor can best be described as an ellipsoidal cap. It is possible for me to measure the length width and height of the tumor, but I have looked in texts that contain common geometric formulas and I can't find the formula for the volume. Could you help? Thanks for your time.
Date: 04/11/2001 at 12:44:21 From: Doctor Rob Subject: Re: Volume of ellispoidal cap Thanks for writing to Ask Dr. Math. The length, width, and height are insufficient to compute the volume. Three parameters, a, b, and c, are necessary to specify the shape of the ellipsoid, and another, r, to specify the plane of the base. You have only three measurements, which would give you three equations in the four parameters. These are insufficient to determine the values of the parameters. Set up a rectangular or Cartesian coordinate system with origin at the center of the ellipsoid, and x-, y-, and z-axes each along one of the axes of the ellipsoid and so perpendicular to each other. Then the equation of the ellipsoid is x^2/a^2 + y^2/b^2 + z^2/c^2 = 1, a, b, c > 0. Let the cutting plane be x/a = r, -1 <= r <= 1, perpendicular to the x-axis. Then the cap bounded by the plane and the surface of the ellipsoid, with r <= x/a <= 1, has volume V, where a b*sqrt(1-x^2/a^2) c*sqrt(1-x^2/a^2-y^2/b^2) V = INT INT INT dz dy dx, r*a -b*sqrt(1-x^2/a^2) -c*sqrt(1-x^2/a^2-y^2/b^2) a b*sqrt(1-x^2/a^2) = 2*c*INT INT sqrt(1-x^2/a^2-y^2/b^2) dy dx, r*a -b*sqrt(1-x^2/a^2) a = 4*Pi*b*c*INT (1-x^2/a^2) dx, r*a V = Pi*a*b*c*(1-r)^2*(2+r)/3, -1 <= r <= 1. Perhaps you can understand why this formula does not appear in the standard collections. The height is a*(1-r); the length and width are b*sqrt(1-r^2) and c*sqrt(1-r^2). This is not enough to determine a, b, c, and r. It is enough to specify the elliptical base, and the vertex of the ellipsoid. You need another measurement, such as the length or width at half the height, to get enough information to solve for a, b, c, and r, and then compute the volume. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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