Current weather in New England makes it hard to not think about snow. So here are two snowball inspired math problems. We welcome your solutions (send to girlsangle “at” gmail.com). Perhaps we’ll post solutions when the last traces of snow have cleared from Boston streets.
Both problems involve a snowball in the shape of a perfect sphere. Mark a point on the surface of the snowball. The snowball rolls without slipping. Also, there is no unnecessary rotation. (In other words, the snowball rotates only around an axis perpendicular to the plane of the great circle that is tangent to the path that the snowball travels.) In each problem, the snowball starts with the marked point touching the surface it is rolling on.
Snowball Problem 1. If the snowball rolls about on a plane, what is the locus of all the points in space that the marked point can visit?
Snowball Problem 2. If the snowball rolls about on another sphere of the same radius, what is the locus of all the points in space that the marked point can visit?