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:
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
We can now express a fundamental property of parallel surfaces, namely that a parallel surface fails to be regular if and only if
Note that we are justified in writing H2-K because
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.
Descirbe the parallel surfaces of a sphere of radius R . For which values of r will the parallel surface at a distance r have singularities? What is the nature of the singularities?
Same question for the hyperbolic paraboloid.
Same question for a circular cylinder, or. more generally, an elliptical cylinder.
Same question for a paraboloid, ellipsoid, or hyperboloid of revolution.
Same question (much more difficult) for a general paraboloid, ellipsoid, or
hyperboloid.