November 1997 Presentation Topic (continued)

Some Tools to Solve the Problems

are, respectively, tangent and normal to S at the point (x,f(x)). See figure 5.

A parametrization of a curve is a map from an interval onto given by

The curve S is said to be smooth at a point if and are smooth at and the tangent vector does not vanish at i.e.,

When (*) fails, the curve S has a singularity at p.

Example: The curve has a singularity at (0,0). Indeed, the tangent vector is given by which vanishes at t=0. If we set and we see that this curve is given by This curve is called a cubic parabola, and this type of singularity a cusp. The graph of this curve can be seen in figure 6.

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