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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