# How to interpret this task?

I have a task given to me in my homework I can not figure out what asks of me. The task is worded like this:

A curve in a plane is given by

$$x(t) = 3(t - \sin(t))$$

$$y(t) = 3(1 - \cos(t))$$

Find the parametric normal-curve $x(s, t), y(s, t)$ through $(x(t), y(t))$, where $s$ is the normal's parameter.

I am not interested in a solution to this task, just an interpretation of what it means so that I will be able to solve it myself. Thanks in advance =)

Consider an arbitrary point $(x_t,y_t)$ on the curve. There is a unique line that passes through that point and is normal (perpendicular) to the curve. Find a parametrization $(u_t(s),v_t(s))$ of that normal line. Now your parametric normal curve will be
$$x(s,t) = u_t(s),\\ y(s,t) = v_t(s).$$
However, this interpretation fails at the cusps of the curve. For example, at $(6\pi,0)=(x(2\pi),y(2\pi))$ there is no normal line.
• So what you say is that it asks me to find a general parametric equation that for every point on the curve $t$ gives me a normal-line through that point by varying the parameter $s$? – Andreas Hagen Mar 24 '13 at 17:18