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Cone Section

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Given a double cone

$\displaystyle K:\ (x-p)^{\operatorname t}v = \pm\cos(\alpha)\,\vert x-p\vert\vert v\vert $

with apex $ p$ ($ p_3\ne 0$), direction $ v$ and semi-apex angle $ \alpha$. The intersection of this cone with plane $ E:\ x_3=0$ is a quadratic curve

$\displaystyle K\cap E:\ a_{1,1} x_1^2 + 2a_{1,2} x_1x_2 + a_{2,2} x_2^2 + 2b_1 x_1 + 2b_2 x_2 + c = 0\,. $

The type of the cone section depends on angle $ \beta$ beween plane $ E$ and line $ p+tv$, $ t\in\mathbb{R}$. We obtain a/an

(Authors: App/Burkhardt/Höllig)
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  automatically generated 3/18/2005