Wednesday, March 9, 2011

153.

The hyperbolic paraboloid is a doubly ruled surface shaped like a saddle. In a cartesian coordinate system, a hyperbolic paraboloid can be represented by the equation

z = \frac{x^2}{a^2} - \frac{y^2}{b^2}.

This is a hyperbolic paraboloid that opens up along the x-axis and down along the y-axis.

The hyperbolic paraboloid, when parametrized as

\vec \sigma (u,v) = \left(u, v, {u^2 \over a^2} - {v^2 \over b^2}\right)

has Gaussian curvature

K(u,v) = {-4 \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^2}

and mean curvature

H(u,v) = {-a^2 + b^2 - {4 u^2 \over a^2} + {4 v^2 \over b^2} \over a^2 b^2 \left(1 + {4 u^2 \over a^4} + {4 v^2 \over b^4}\right)^{3/2}}.


An image of the structure is found below.




A very famous item takes the shape of the structure described above. Identify.

Posted by at
Labels:
Subscribe to: Post Comments (Atom)