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 =)
This is how I interpreted the problem:
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.