Paraboloid of revolution

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

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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 : .