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Henrik Holst edited Ellipsens_ekvation_E_x_frac__.tex
almost 9 years ago
Commit id: 4e5aa6d9152cf886a78056808a197ebee10b4871
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Ellipsens ekvation: The equation for the ellipsoid:
$$E(x) := \frac{x_1^2}{e_1^2} + \frac{x_2^2}{e_2^2} + \frac{x_3^2}{e_3^2} - 1 = 0$$
Vi har en punkt Let $y$
utanför ellipsoiden: be a point outside the ellipsoid:
$$y = \begin{bmatrix} y_1 & y_2 & y_3 \end{bmatrix}$$
Vi har för en punkt For any point $x$
på ellipsen: on the ellipsoid we have:
$$y - x = \frac{1}{2} t \nabla E(x) = t \begin{bmatrix} \frac{x_1}{e_1^2} & \frac{x_2}{e_2^2} & \frac{x_3}{e_3^2} \end{bmatrix}$$
eller ekvivalent formulerat or equivalently,
$$y_k = \left( 1 + \frac{t}{e_k^2} \right) x_k, \qquad k=1,2,3.$$
We can formulate this as
$$\frac{e_k^2}{t+e_k^2} y_k = x_k, \qquad k=1,2,3$$
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