The class notes yesterday should have included some discussion of the “self-intersecting curve” problem. Here it is a day late. Suppose x = x(t) and y = y(t) are given; a point of self-intersection is said to occur when, for two different values of the parameter— and , say—one has the same point of ; in other words and .
In the example at hand, we have and . Assume that the parameter values and give a point of self-intersection for this curve. From the equation for x we see that then ; since we are assuming thqt , this implies that . Substituing in the equation for y gives
; factoring the LHS gives , the opposite of the RHS. Since a number equal to its own opposite is zero, we have RHS = 0; factoring gives ; since (because ), this implies that .
So our parametric curve intersects itself for the t values and . One easily obtains the desired values of to complete the exercise.