I have the given paragraph in the book I am reading on knot theory:
It is essential to distinguish a path $a$ from the set of image points $a(t)$ in $X$ visited during the interval $[0, ||a||]$. Different paths may very well have the same set of image points. For example, let $X$ be the unit circle in the plane, given in polar coordinates as the set of all pairs $(r, \theta)$ such that $r=1.$ The two paths $$a(t) = (1,t), 0 \leq t \leq 2\pi,$$ $$b(t) = (1,2t), 0 \leq t \leq 2\pi,$$ are distinct even though they have the same stopping time, same initial and terminal point, and same set of image points.
But I do not understand how those two paths have the same stopping time? stopping time of the interval $[0, ||a||]$, is defined as $||a||$ and it is assumed that $0 \leq ||a||$. what is $||a||$ & $||b||$ in our case?
Also, I do not understand how they have the same terminal point, and same set of image points. Could anyone explain this for me please?
Both of them start and end at $(1,0)$ as the angle is calculated mod $2 \pi$ and thus they have the same terminal point. Both of these loops correspond to going around the unit circle, but $b$ does that twice. This means the image points are both times given by the entire unit circle. In case you know how to multiply loops, then you can see that $b = a^2$.