What is the equation of a 3D cone with generalised tilt?

Question

What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space

Answer 1

Equation of circular cone with singular point at origin \begin{align*} \begin{pmatrix} x & y & z \end{pmatrix} \begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} &= 0 \\ ax^2+by^2+cz^2+2(fyz+gzx+hxy) &=0 \end{align*}

where $\begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix}$ has eigenvalues in form of $\pm \lambda^2$, $\pm \lambda^2$ and $\mp \mu^2$.