9.2: Normal Images of Tubes around Smooth Curves on Surfaces

The object which we will consider first is a composite of several others. We take the positive and negative parallel surfaces to a patch P on a surface and join them with the normal tube at the same distance to the boundary curve of the patch.

For a closed smooth space curve X(t) on a surface X(u,v), we may describe the tube of radius r around the curve in terms of the unit normal N(t) to the surface at the point X(t) and the unit vector U(t) = N(t) × T(t). We define the tube as

Y^r(t,v) = X(t) + rcos(v)U(t) + rsin(v)N(t)

X^r(u,v) = X(u,v) ± rN(u,v)

Notice that this is a compact surface.

Demonstration 1: Closed Parallel Surfaces ("Moldy Potato Chip")

In this demonstration, we draw a yellow circle in the domain whose center and radius can be changed using the two red hotspots. This circle is mapped to a curve X(t) on the surface, which is a torus by default. The closed curve X(t) on the torus encloses a region P that resembles a potato chip to some extent. In the surface window, we draw the closed parallel surface at a distance r from the "chip". It is sometimes helpful to think of the closed parallel surface as a "moldy potato chip", where hairs of length r grow normal to the surface of the chip. On the edge of the potato chip, along the curve X(t), the hairs extend radially outward to create part of a tube surface. Together, the parallel surfaces to the chip and the tube surface create a closed surface that is topologically equivalent to a sphere. Consequently, we expect the Gauss map of such a surface to cover the unit sphere exactly once. In the demo, the tube surface Y^r(t,v) is colored green and the parallel surfaces to the interior are colored purple. Regions of positive and negative Gaussian curvature are indicated by lighter and darker shades of the two colors respectively. The color-coding also reveals how the parallel surfaces are taken onto the unit sphere through the Gauss map. This allows us to verify that the sphere is, in fact, covered exactly once.

If the original patch is simply-connected and closed, the closed parallel surface S will be topologically equivalent to a sphere. This means that the Gauss map algebraically covers the sphere once. Analytically, this geometric consideration can be written as

_SΚ^*dA = 4π

As the demo illustrates, this parallel surface is made up of two parts: (1) the circular half-tube around the curve and (2) the part that is parallel strictly to the interior of the patch. An important fact is

₍₁₎Κ^*dA = 2_∂Pκ_gds

_SΚ^*dA = ₍₁₎Κ^*dA + ₍₂₎Κ^*dA,     or
_SΚ^*dA = 2_∂Pκ_gds + 2_PΚdA            

_PΚdA + _∂Pκ_gds = 2π

Exercises

1. For various sizes of ellipsoids, what happens to these quantities (the TGC, the TGeoC and the Sum) as we change the curve?

2. Smaller curves imply better calculation precision. What happens to the quantities and especially the Sum, as you scale the size of the curve down? (For the ellipse, both a and b need to be scaled, while for the egg shape, only b needs scaling.)

3. More technical: We know that the sum of the quantities calculated should be equal to 2π. The default has TGC < TGeoC. What changes can be done to make TGC > TGeoC or even TGC = TGeoC ? Why?