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.

 Solar oven of Odeillo in the Pyrenees. See the principle at parabola. Parabolic antenna

© Robert FERRÉOL, Jacques MANDONNET 2017