info prev up next book cdrom email home

Monge Patch

A Monge patch is a Patch ${\bf x}:U\to\Bbb{R}^3$ of the form

{\bf x}(u,v)=(u, v, h(u,v)),
\end{displaymath} (1)

where $U$ is an Open Set in $\Bbb{R}^2$ and $h:U\to\Bbb{R}$ is a differentiable function. The coefficients of the first Fundamental Form are given by
$\displaystyle E$ $\textstyle =$ $\displaystyle 1+{h_u}^2$ (2)
$\displaystyle F$ $\textstyle =$ $\displaystyle h_uh_v$ (3)
$\displaystyle G$ $\textstyle =$ $\displaystyle 1+{h_v}^2$ (4)

and the second Fundamental Form by
$\displaystyle e$ $\textstyle =$ $\displaystyle {h_{uu}\over\sqrt{1+{h_u}^2+{h_v}^2}}$ (5)
$\displaystyle f$ $\textstyle =$ $\displaystyle {h_{uv}\over\sqrt{1+{h_u}^2+{h_v}^2}}$ (6)
$\displaystyle g$ $\textstyle =$ $\displaystyle {g_{vv}\over\sqrt{1+{h_u}^2+{h_v}^2}}.$ (7)

For a Monge patch, the Gaussian Curvature and Mean Curvature are
$\displaystyle K$ $\textstyle =$ $\displaystyle {h_{uu}h_{vv}-{h_{uv}}^2\over(1+{h_u}^2+{h_v}^2)^2}$ (8)
$\displaystyle H$ $\textstyle =$ $\displaystyle {(1+{h_v}^2)h_{uu}-2h_u h_v h_{uv}+(1+{h_u}^2)h_{vv}\over (1+{h_u}^2+{h_v}^2)^{3/2}}.$ (9)

See also Monge's Form, Patch


Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 305-306, 1993.

© 1996-9 Eric W. Weisstein