3.5.1 Explicit surfaces

Very often a surface is given by an explicit form . It is, therefore, convenient to have analytic equations for the Gaussian and mean curvatures expressed in terms of the derivatives of the height function . As we mentioned in Sect. 1.1 the explicit form can be converted into a parametric form where and . This form is often referred to as Monge form , and the surface is called a Monge patch. It is straightforward to evaluate

and hence

Example 3.5.1. Let us compute the Gaussian and mean curvatures of the hyperbolic paraboloid (in Example 3.2.1 we used its parametric form) using the explicit formulae (3.63) to (3.67). Since

we have

and hence

Here we can observe that the Gaussian curvature is always negative and thus all the points on a hyperbolic paraboloid are hyperbolic points. Furthermore, since and , the surface intersects its tangent plane at the iso-parametric lines (see Sect. 3.3 last paragraph). Also from (3.49) and (3.50) we obtain

where it is very easy to show that and for all .