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if i have an ellipsoid and a plane oriented in any way in a 3 dimensional coordinate system, and they intersect; is there a way to find an equation that describes (or at least approximates) all points of intersection for example, an ellipsoid that has twice the diameter between the "poles" than the "equator" will yield a different 2 dimensional ellipse on the plane that cuts through depending how the plane is oriented in space and how it cuts through the 3d ellipsoid; however, the points of intersection will form some kind of a 2 dimensional ellipse on the plane and im just wondering how can i find that should i be using the normal vector form for the plane or can i use a general x,y,z implicit form (or explicit form) any guidance would be much appreciated :)

thank you

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Pick a point $\mathbf p$ on the plane and two linearly independent vectors $\mathbf e_1$ and $\mathbf e_2$ lying along it. Then any point on the plane is of the form $\mathbf x = \mathbf p + u\mathbf e_1 + v\mathbf e_2$ for real numbers $u$ and $v$. Plug this into the equation of the ellipsoid and you get a quadratic in $u$ and $v$; this defines your ellipse of intersection. –  Rahul Jan 13 at 21:00

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Take a look at this paper http://www.scirp.org/journal/PaperInformation.aspx?PaperID=24506 I believe it has what your looking for unfortunately its pretty dense.

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