November 1997 Presentation Topic (continued)

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

t > F(t) = (F_1(t), F_2(t)).

(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

Navigation Panel:

Go backward to Problems

Go up to Introduction and Contents

Go forward to Solutions to the problems

Switch to graphical version (better pictures & formulas)

Access printed version in PostScript format (requires PostScript printer)

Go to SIMMER Home Page

Go to The Fields Institute Home Page

Go to University of Toronto Mathematics Network Home Page