= Solution
The distance from the origin is $r=4\pi-t$. The direction angle, measured counterclockwise from the positive horizontal axis, is $\theta=\pi/2-t$ modulo $2\pi$. Thus the path is an <Archimedean spiral>, turning clockwise through two complete revolutions as its <radius> decreases linearly from $4\pi$ to zero.
It starts at $(0,4\pi)$, reaches $(7\pi/2,0)$ at $t=\pi/2$, crosses $(0,-3\pi)$ at $t=\pi$, returns to the positive vertical axis at $(0,2\pi)$ when $t=2\pi$, and ends at the origin when $t=4\pi$. These crossings fix both the scale and direction of the sketch.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1-inward-spiral.png]
{title=Two clockwise inward turns of the spiral, with time-direction arrows and selected axis crossings}
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