deletions | additions       diff --git a/section_Equations_E_x_frac__.tex b/section_Equations_E_x_frac__.tex  index d196482..43b80b7 100644  --- a/section_Equations_E_x_frac__.tex  +++ b/section_Equations_E_x_frac__.tex 
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$$E(x) := \frac{x_1^2}{e_1^2} + \frac{x_2^2}{e_2^2} + \frac{x_3^2}{e_3^2} - 1 = 0$$  Let $y = \begin{bmatrix} y_1 & y_2 & y_3 \end{bmatrix}^T$ be a point outside the ellipsoid.  For any point $x = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}^{T}$ on the ellipsoid we have\footnote{http://www.geometrictools.com/Documentation/DistancePointEllipseEllipsoid.pdf}:  $$y - x = \frac{1}{2} t \nabla E(x) =  t \begin{bmatrix}   \frac{1}{e_1^2} && \\   & \frac{1}{e_2^2} & \\   && \frac{1}{e_3^2}   \end{bmatrix} x  $$ E(x),$$  or equivalently, equivalently in component form,  $$y_k = \left( 1 + \frac{t}{e_k^2} \right) x_k, \qquad k=1,2,3.$$  The values $x_k$ can be expressed as a function of $t$:  $$x_k(t):=\frac{e_k^2}{t+e_k^2} y_k, \qquad k=1,2,3$$ 
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