Monge Patch
A Monge patch is a patch x:U->R^3 of the form
| x(u,v)=(u,v,h(u,v)), |
(1)
|
where U is an open set in R^2 and h:U->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 = [画像:(h_(uu))/(sqrt(1+h_u^2+h_v^2))]
(5)
f = [画像:(h_(uv))/(sqrt(1+h_u^2+h_v^2))]
(6)
g = [画像:(h_(vv))/(sqrt(1+h_u^2+h_v^2)).]
(7)
For a Monge patch, the Gaussian curvature and mean curvature are
See also
Monge's Form, PatchExplore with Wolfram|Alpha
WolframAlpha
More things to try:
References
Gray, A. "A Monge Patch." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 398-401, 1997.Referenced on Wolfram|Alpha
Monge PatchCite this as:
Weisstein, Eric W. "Monge Patch." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MongePatch.html