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CONE
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