9.3: Normal Images of Tubes around Piecewise Smooth Curves

Consider a region on a surface which is bounded by a number of smooth curves meeting at vertex points where the limit of the unit tangent vectors from above and below both exist. Such a curve is called a piecewise smooth curve. If the limit tangent vector from below makes an angle α with the limit tangent vector from the next, we say that the exterior angle at the vertex is α. For example, since the total curvature of straight line segments in the plane is zero, there will be some exterior angle at the vertex for any polygon. If the polygon is regular with n sides, then each exterior angle will be α = 2π/n.

In the previous section, we obtiained a relationship between the total Gaussian curvature of a smooth portion of surface and the total geodesic curvature of its smooth boundary. To generalize it to piecewise smooth curves, the relation cannot hold as is.

Demonstration 3: Polygons onto Surfaces ("Moldy Frito")

In this demonstration, we draw a yellow square in the domain whose center and side length can be changed using the two red hotspots. This square is mapped to the piecewise continuous 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 frito. In the surface window, we draw the closed parallel surface at a distance r from the "frito". As in the "moldy potato chip" demo, it may help to think of the surface in this demo as a "moldy frito", where hairs of length r grow normal to the frito. For the interior region of the frito, the normal hairs create the two parallel surfaces to the torus over the square domain. Along the edge of the frito (excluding the corners), the hairs extend radially outward from the curve X(t) to create part of a tube surface. And at the corners, the hairs extend radially outward from the individual points to create parts of a sphere. Together, the parallel surfaces to the frito, the tube surface, and the corner surfaces create a closed surface that is topologically equivalent to a sphere. Consequently, we expect the Gauss map to cover the unit sphere exactly once. In the demo, the tube surface is colored green, the parallel surface surface is colored purple, and the spherical lunes are colored orange. Lighter and darker shades of the same color correspond to regions of positive and negative Gaussian curvature 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.


The Gauss-Bonnet Formula for Piecewise Smooth Curves on Regular Surfaces

Let S be the closed parallel surface around a patch P such that P is simply-connected and closed. Then, as in the case of the moldy frito, S is topologically equivalent to a sphere. Consequently, the Gauss map of S covers the unit sphere once, which means that

SΚ*dA = 4π

We saw in the demonstration that the closed parallel surface is made up of three parts (1) the circular half-tube around the curve (excluding the corners), (2) the part that is parallel strictly to the interior of the patch, and (3) the spherical lunes around the corners. From the previous section, we showed that

(1)Κ*dA = 2∂Pκgds

where the integral of geodesic curvature can be split up into separate integrals corresponding to each of the piecewise connected curves that make up the boundary X(t). We also note that the total curvature of the two surfaces parallel to the interior of the patch is equal to twice the total curvature over the patch itself. That is,

(2)Κ*dA = 2PΚdA

All that is left to do is to find an expression for the total curvature for the spherical lunes located at the vertices of the patch. Using the demo, observe that the tube of radius r about a piecwise smooth curve will have gaps at each vertex. The ends of the circular half-tubes will be semicircles that meet at antipodal points. The angle α between the two semicircles at the poles will be equal to the exterior angle of the vertex. Thus the surface connecting the half-tubes at a distance r from a vertex with exterior angle α will be a spherical lune with angle α. We leave it as an exercise to show that the area of the normal image of a lune with angle α on a sphere of radius r is equal to 2α (independent of r). Suppose we have a piecewise smooth curve X(t) with n vertices. Let α1, α2, ..., αn be the exterior angles of X(t). Then there are n spherical lunes connecting the tube surfaces. The total curvature of these lunes is equal to the sum of the areas of all of their the normal images. This means that

(3)Κ*dA = 2∑ αi

With the above three equations, we can now expression the total Gaussian curvature of the surface S in terms of the total geodesic curvature of the boundary curve X(t), the total Gaussian curvature of the patch P, and the exterior angles αi.

SΚ*dA = (1)Κ*dA +(2)Κ*dA + (3)Κ*dA
SΚ*dA = 2∂Pκgds + 2PΚdA + 2∑ αi           

From this last equation, we derive the Gauss-Bonnet Formula for piecewise smooth curves on a regular surface:

PΚdA + ∂Pκgds + ∑ αi = 2π

Exercises

    1. What can be said of the sum of exterior angles for any region that has Gaussian curvature strictly positive? Same question with a region of strictly negative Gaussian curvature. Note: Again, these are not polygons of the surface.

    2. The interior angle is defined as π - α where α is the exterior angle. The demo will not necessarily illustrate this. What can be said about the sum of interior angles for surface polygons in strictly positive Gaussian curvature? negative Gaussian curvature?

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