deletions | additions       diff --git a/The_equation_for_the_ellipsoid__.tex b/The_equation_for_the_ellipsoid__.tex  index 79a193d..91f2e00 100644  --- a/The_equation_for_the_ellipsoid__.tex  +++ b/The_equation_for_the_ellipsoid__.tex 
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The equation for the ellipsoid:  $$E(x) := \frac{x_1^2}{e_1^2} + \frac{x_2^2}{e_2^2} + \frac{x_3^2}{e_3^2} - 1 = 0$$  Let $y$ be a point outside the ellipsoid:  $$y = \begin{bmatrix} y_1 & \\  y_2 & \\  y_3 \end{bmatrix}$$ For any point $x$ on the ellipsoid we have:  $$y - x = \frac{1}{2} t \nabla E(x) = t \begin{bmatrix} \frac{x_1}{e_1^2} \frac{1 \begin{bmat}{e_1^2} && \\  & \frac{x_2}{e_2^2} \frac{1}{e_2^2}  & \frac{x_3}{e_3^2} \end{bmatrix}$$ \\   && \frac{1}{e_3^2}   \end{bmatrix} \begin{bmatrix}  x_1 \\ x_2 \\ x_3  \end{bmatrix}  or equivalently,   $$y_k = \left( 1 + \frac{t}{e_k^2} \right) x_k, \qquad k=1,2,3.$$  We can formulate this as 
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