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|Cartesian equation: ,
Cartesian parametrization: .
Parametrization the coordinate lines of which are the curvature lines (case ): (figure opposite).
Total curvature: .
The projections on xOy of the curvature lines are the ellipses , and the hyperbolas , with the general formula: .
|Volume of the paraboloidal bowl with height h, the semi-axes of the ellipse at the summit being a and b (): (half of the circumscribed cylinder).||
The elliptic paraboloids can be defined as the surfaces generated by the translation of a parabola (here with parameter p) along a parabola in the same direction (here with parameter q) (they are therefore translation surfaces).
The sections by vertical planes are parabolas and the sections by horizontal planes are ellipses.
||View of one of the two families of circles included in any elliptic paraboloid, even if it is not of revolution, with the corresponding umbilic.|
The paraboloid of revolution (or circular paraboloid) corresponds to the case p = q.
See at hyperbolic paraboloid a boxed text about confocal paraboloids.
Bird nest shaped like a paraboloid. If you have something better...
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© Robert FERRÉOL 2017