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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 parametrizations : |
where coordinates lines are parallel circles and meridian
parabolas :
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where coordinates lines are parabolas :
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other parabolas :
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ellipses :
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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. |
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Volume of the paraboloidic bowl with height h,
the radius of the circle at the summit being R ( Area of this bowl: |
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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 parameterization: Area for |
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![]() Solar oven of Odeillo in the Pyrenees. See the principle at parabola. |
![]() Parabolic antenna |
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© Robert FERRÉOL, Jacques MANDONNET 2017