Cartesian equation of a cone with vertex O: f(x, y, z) = 0 with f homogeneous. In particular: z = f(x, y) with f homogeneous of degree 1. Cartesian parametrization:  (directrix  ). Cylindrical equation:  (directrix ). Parametrization stemming from the polar coordinates  of the plane of development of the cone:    with . Parametrization with geodesics (other than the generatrices): .

A cone (or conical surface) is a ruled surface the generatrices of which pass through a fixed point O (its vertex), in other words, a surface globally invariant under any homothety centered on O (with ratio 0).
A curve traced on the cone that intersects all the generatrices is called a directrix of the cone; there exists a unique cone with given vertex and directrix.

An algebraic surface with equation f(x,y,z) = 0 is a cone with vertex O if and only if the polynomial f is homogeneous. The degree of f is then the degree of the cone (as an algebraic surface).
The sections of this cone by planes that do not pass by O are then the various curves (projectively equivalent) with homogeneous equation .

Examples:
    - cone of revolution
    - elliptic cone
    - sinusoidal cone
    - Cartan's umbrella

Compare to the conoids.
 

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© Robert FERRÉOL   2017