The Paraboloid

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Introduction:

The paraboloid is so called because it has parabolic cross sections (see right). The plot above and the parameterization below describe a circular paraboloid, because cross-sections parallel to the xy plane are circular. It is also possible to define an elliptic paraboloid whose cross-sections are elliptic.


Definition:

Properties:

Partial Derivatives:

Because z is a function of x and y, we can take partial derivatives:

     

     

Tangent Planes:

At u = u0, v = v0, the tangent plane to the surface is parameterized by:

Infinitesimal Area:

The infinitesimal area of a patch on the surface is given by

Gaussian Curvature:

Gaussian curvature of the surface.
Surface colored by Gaussian curvature.

Mean Curvature:

Mean curvature of the surface.
Surface colored by Mean curvature.