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 parametrizations :
 where coordinates lines are parallel circles and meridian parabolas : (). where coordinates lines are parabolas : other parabolas : ellipses :
 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:  ; approximation for a small h  : .

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

© Robert FERRÉOL, Jacques MANDONNET 2017