I found the following problem online:
"The white seam on a tennis ball is the gap between two green pieces of fabric which form the fuzzy covering of the ball. Ever wonder what 3-dimensional curve the seam follows?
Consider two sets of points A and B, where A is a circular arc running from the North Pole to the South Pole of the tennis ball, along the Prime Meridian (as if the ball were the Earth), and B is a circular arc running along the equator from 90 degrees West longitude to 90 degrees East longitude, going the other way around to avoid the Prime Meridian. Let C be the set of points whose minimum distance to a point of A equals the minimum distance to a point of B. Calculate the ratio of the length of C over the circumference of the tennis ball, rounded to the nearest thousandth."
I tried but to solve it but I think I am interpreting it wrong visually. Does anyone have a solution? Thanks in advance!
Best Answer
When all else fails, draw a diagram.
The four diagrams show four different orientations of the tennis ball, where each diagram is obtained from the last by rotating 90° downwards towards you (hence the distinguished endpoints of the arcs A and B). It is easy to see that C, the tennis ball's seam (the U in each diagram), breaks into four identical curves, which are the set of points equidistant between one arc and the endpoint of the other arc.
These arcs are semicircles, and their length is half that of the small circle at 45° N/S latitude. If the tennis ball's radius is 1, the radius of the small circle is $\frac{\sqrt2}2$ and the arc length of each semicircle $\frac{\pi\sqrt2}2$, which means that the seam's total length is $2\pi\sqrt2$. The tennis ball's circumference is $2\pi$, so the ratio asked for in the original question is $\frac{2\pi\sqrt2}{2\pi}=\sqrt2\approx 1.414$.