Parallel Surfaces and Mean Curvature

The concept of a parallel surface to a smooth surface in three-dimensional space is analogous to that of a parallel curve to a smooth curve in the plane. In the plane, we saw that the properties of parallel curves were closely related to the notion of curvature; it is the same for surfaces in three-space. For a surface X(u,v) we define the parallel surface at distance r by

The parallel surface Xr(u,v) will be regular at all points where the partial derivative vectors are not linearly dependent, i.e. when Xru( u,v)·X rv(u,v) is not the zero vector.

We may compute the partial derivatives as follows:

and similarly The cross product of these two vectors is then

We have already defined Nu(u,v) ·Nv( u,v)=K(u,v)Xu(u,v) ·Xv( u,v) . Since Xu(u,v) ·Nv( u,v) is the cross product of two vectors in the tangent plane, it also is a multiple of Xu(u,v) ·Xv( u,v) and so is Xv(u,v) ·Nu( u,v). We may then define another scalar function H(u,v) over the domain of the surface by the condition

The scalar function H(u,v) , for reasons that will become clear very soon, is called the mean curvature of the surface.

We can now express a fundamental property of parallel surfaces, namely that a parallel surface fails to be regular if and only if

This will occur if and only if r is a root of the quadratic equation i.e. if These two values are called the Radii of Curvature of the surface at the point. The reciprocals of the radii of curvature are called the Principal Curvatures . By an algebraic manipulation, the principal curvatures are given by the formulae Now, the terminology of mean curvature becomes apperent. Using the principal curvatures we have K=k1k2 and H=k1+k22 .

Note that we are justified in writing H2-K because

A point for which k1=k2 is called umbilic.

A point at which the principal curvatures are equal is called an umbilic point. This means that Parallel Surfaces Demo

You may toggle the displays of normals between the surface and the parallel surface. The distance from the original surface to the parallel surface is determined by a slider in control window. In Gauss Map of Parallel Surface the normal surface to the parallel surface is drawn with color scaled between red and green according to the height of the normal along the z-direction. This last display permits an easy detection of singularities in the parallel surface.

This demonstration shows the domain of a surface, the surface, and a single parallel surface at a distance determined by a slider.