General equation of a cone

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What is the general equation of a cone in $\mathbb{R}^3$ space?

There should be no assumptions about the location of the vertex, direction of the axis or aperture angle, these should all be variable.

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1 Answer

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A point X (where $X = [x,y,z]^T$ ) on a cone will satisfy: $$(X-U)^TM(X-U) = 0,$$ where $U$ is the vertex of the cone , $\hat{D}$ is the axis of the cone (unit vector) and, $$ M = \hat{D}\hat{D}^T - \cos^2(\alpha)I_{3 \times 3},$$ where $\alpha$ is the aperture angle of the cone and $I$ is the identity matrix.

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