7.1: Evolute Surfaces

Finally, the last family of surfaces we study are called evolute surfaces to a suface. We define these evolute surfaces in a manner similar to that in which we defined evolutes to plane curves. For surfaces we use the principal curvatures to define evolute surfaces E₁(u,v) and E₂(u,v). We have:

E₁(u,v)=X(u,v) + (1/κ₁(u,v))N(u,v)
E₂(u,v)=X(u,v) + (1/κ₂(u,v))N(u,v)

In this demonstration, the checkered surface is an ellipsoid, whose three axes are determined by the variables a, b, and c. The two evolute surfaces of the ellipsoid are also shown in the surface window, where E₁(u,v) is colored green and E₂(u,v) is colored cyan. In the domain window, there is a color coded square, or patch, that can be moved around and resized using the two white dots (hotspots). The image of this patch appears on the surface X(u,v), on the normal map N(u,v), and on the first evolute surface E₁(u,v). The patch is helpful in showing how the evolute surfaces are derived from the original surface. To create E₁(u,v) in the domain of the patch, we start with the patch on the surface and then from this region we extend vectors with length 1/κ₁(u,v) that are normal to the surface. These vectors project the image of the patch onto the evolute surface.

The display of the evolute surfaces is toggled independently by Evolute1 and Evolute2 . In some smooth surfaces there are singularities that are due only to the choice of parametrization and not to the nature of the surface itself, such the north and south poles of a sphere with the usual latitude-longitude parametrization. In calculating the evolutes, these singularities often can cause problems. If this is the case, it is best to restrict the domain of the surface to avoid the singular points.

This demo will let you input a surface and domain; it then displays the surface, and the evolute surfaces and the Gaussian curvature of the evolute surfaces.

Exercises

1. What are the evolute surfaces of an ellipsoid of revolution? How do they change as the ellipsoid is stretched into an ellipsoid with three unequal axes?

2. Describe the evolute surfaces for a paraboloid of revolution. How do these surfaces change when the paraboloid is stretched into an elliptic paraboloid with unequal axes?

3. What can we say about the Gaussian curvatures of evolute surfaces? What happens in the case of an elliptic hyperpoloid?

4. When are the evolute surfaces to an ellipse singular (besides removable singularities)?