Some Tools to Solve the Problems

Recall that for a curve S = { (x,y) in R^2 : y=f(x) }, the vectors

                     1
        T(x) = ------------------- (1, f'(x))     and
               [1+(f'(x))^2]^(1/2)
        
                     1
        N(x) = ------------------- (-f'(x), 1)
               [1+(f'(x))^2]^(1/2)

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

Figure 5: PICTURE AVAILABLE IN GRAPHICAL VERSION ONLY

A parametrization of a curve S in R^2 is a map from an interval I in R onto S in R^2 given by

F: I -->S in R^2
t > F(t) = (F_1(t), F_2(t)). The curve S is said to be smooth at a point p in S if p=F(t_0), F_1 and F_2 are smooth at t_0 and the tangent vector T(t_0)=(F_1'(t_0), F_2'(t_0)) does not vanish at t=t_0, i.e.,

(F_1'(t_0))^2 + (F_2'(t_0))^2 != 0. (*)

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

Example: The curve F(t)=(t^3,t^2), t in (-1,1) has a singularity at (0,0). Indeed, the tangent vector is given by T(t)=(3t^2,2t), which vanishes at t=0. If we set x=t^3 and y=t^2 we see that this curve is given by { (x,y) : y^3 = x^2 }. 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.

Figure 6: PICTURE AVAILABLE IN GRAPHICAL VERSION ONLY

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