Editing Paraboloid (section)

== Curvature ==
The elliptic paraboloid, parametrized simply as
<math display="block">\vec \sigma(u,v) = \left(u, v, \frac{u^2}{a^2} + \frac{v^2}{b^2}\right) </math>
has [[Gaussian curvature]]
<math display="block">K(u,v) = \frac{4}{a^2 b^2 \left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^2}</math>
and [[mean curvature]]
<math display="block">H(u,v) = \frac{a^2 + b^2 + \frac{4u^2}{a^2} + \frac{4v^2}{b^2}}{a^2 b^2 \sqrt{\left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^3}}</math>
which are both always positive, have their maximum at the origin, become smaller as a point on the surface moves further away from the origin, and tend asymptotically to zero as the said point moves infinitely away from the origin.

The hyperbolic paraboloid,<ref name="Weisstein" /> when parametrized as
<math display="block">\vec \sigma (u,v) = \left(u, v, \frac{u^2}{a^2} - \frac{v^2}{b^2}\right) </math>
has Gaussian curvature
<math display="block">K(u,v) = \frac{-4}{a^2 b^2 \left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^2} </math>
and mean curvature
<math display="block">H(u,v) = \frac{-a^2 + b^2 - \frac{4u^2}{a^2} + \frac{4v^2}{b^2}}{a^2 b^2 \sqrt{\left(1 + \frac{4u^2}{a^4} + \frac{4v^2}{b^4}\right)^3}}. </math>