Surfaces Associated with Plane Curves

In this section we will discuss flat surfaces, that is, surfaces that lie in a plane. Such surfaces are sometimes referred to as "areas". We will say for the moment that a point on a surface is regular if the normal exists at that point. If a point is not regular it is called a singular point.

For a plane curve we have already encountered the concept of a parallel curve,

X_r(t )=X( t)+rU( t)

Another important strip in the plane is the flat cone from the origin over a curve X(t) . This may be described as the collection of vectors rX(t) where t runs over the domain of the curve and r runs from 0 to 1 . In the case where X(t) describes a convex curve traversed in a counterclockwise way with the origin in its interior, the strip covers the convex region exactly once and the area of the strip is the area of the region. This is also true for a star-shaped region, so that two distinct segments from the origin to a point of the curve intersect only at the origin.

A third type of flat surface is one generated by taking the set of tangent lines to a plane curve. This is called a tangential surface.

Demonstration 3: Flat Surfaces from Plane Curves

Java not enabled.

dtext4a-3.gif">-->

In this demo, you specify the coefficients Ct and Ut of T and U, respectively, as functions of t and u. (t and u, the intervals over which these functions are evaluated, aren't the same as T and U, the unit tangent and unit normal vectors.) You make make use of Kg(t), the curvature of the curve at t.

This demo displays an area in the plane that is generated from a plane curve.

In the default curve and surface in the above demo, where does the surface have singularities? Why? What happens to the parallel strip if the curve has inflection points?

What will cause the flat tangential surface to have singularities? What happens if the curve has inflection points?