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PARABOLOID OF REVOLUTION
The paraboloid of revolution is the surface obtained by
the revolution of a parabola around its axis.
Cylindrical equation: .
Cartesian equation: . Quadric. Cartesian parametrization: () First fundamental quadratic form: . Surface element: . Second fundamental quadratic form: Total curvature: All the points are elliptic and there is an umbilic: the vertex O. 

Volume of the paraboloidic bowl with height h,
the radius of the circle at the summit being R ():
(half of the circumscribed cylinder).
Area of this bowl: . 
Other parametrization: (the coordinate lines are ellipses). 

Remarkable curves traced on the paraboloid of revolution:
 the curvature lines are the parallels (circles)
and the meridians (parabolas),
 there are no asymptotic lines,
 the geodesics
are the curves solution of: ,
 the helices.
 the 3D
basins.
A paraboloid of revolution can be physically obtained by rotating a liquid at constant speed around an axis. 
See, more generally, elliptic
paraboloids.
The parabola can also be rotated around a line perpendicular
to its axis.
We obtain a surface that looks like a hyperboloid of revolution, but which is of degree 4. Cylindrical equation: , cartesian equation: . Cartesian parameterization:. Area for : . 
Solar oven of Odeillo in the Pyrenees. See the principle at parabola. 
Parabolic antenna 
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© Robert FERRÉOL, Jacques MANDONNET 2017