Monge Patch

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

$${\bf x}(u,v)=(u, v, h(u,v)),$$ (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

$$E = 1+{h_u}^2$$ (2)

$$F = h_uh_v$$ (3)

$$G = 1+{h_v}^2$$ (4)

and the second Fundamental Form by

$$e = {h_{uu}\over\sqrt{1+{h_u}^2+{h_v}^2}}$$ (5)

$$f = {h_{uv}\over\sqrt{1+{h_u}^2+{h_v}^2}}$$ (6)

$$g = {g_{vv}\over\sqrt{1+{h_u}^2+{h_v}^2}}.$$ (7)

For a Monge patch, the Gaussian Curvature and Mean Curvature are

$$K = {h_{uu}h_{vv}-{h_{uv}}^2\over(1+{h_u}^2+{h_v}^2)^2}$$ (8)

$$H = {(1+{h_v}^2)h_{uu}-2h_u h_v h_{uv}+(1+{h_u}^2)h_{vv}\over (1+{h_u}^2+{h_v}^2)^{3/2}}.$$ (9)

See also Monge's Form, Patch

References

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