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Math Help - ellipsoid

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    ellipsoid

    What is the external normal of an ellipsoid at a certain point (x,y,z)? How do I compute it?

    Thanks,
    Yuval
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    Quote Originally Posted by yuvalroc View Post
    What is the external normal of an ellipsoid at a certain point (x,y,z)? How do I compute it?

    Thanks,
    Yuval
    Let \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 be your ellipsoid. Define the function F(x,y,z) = \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} - 1.
    Now use the property that \nabla F(x_0,y_0,z_0) is orthogonal to the ellipsoid F(x,y,z)=0.
    Thus, \left( \frac{2x_0}{a}, \frac{2y_0}{b}, \frac{2z_0}{c} \right) is orthogonal to the ellipsoid.
    It remains to put a \pm 1 in front of it if you want it to be internal or external.
    Put the right sign and you have your outward normal vector.
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