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November 1997 Presentation Topic (continued)

Some Tools to Solve the Problems

Recall that for a curve  (IMAGE) the vectors


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


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


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


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

Example: The curve  (IMAGE) has a singularity at (0,0). Indeed, the tangent vector is given by  (IMAGE) which vanishes at t=0. If we set  (IMAGE) and  (IMAGE) we see that this curve is given by  (IMAGE) 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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