November 1997 Presentation Topic (continued)

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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