A Rotating Reuleaux Triangle
A Reuleaux triangle is a curve of constant width formed from an equilateral triangle by joining each pair of vertices with a circular arc centered at the third vertex (each radius is equal to the side length of the triangle). A Reuleaux triangle can be rotated inside a square and very nearly passes over every point of the square at some point in its rotation. This geometry is the basis for a drill bit that makes square holes, which was first patented by Harry Watts in 1914. The envelope of the Reuleaux triangle's rotation (blue) is elliptical at the corners and contains nearly 99% of the area of the square. The path of the centroid (purple) appears circular but in fact consists of an elliptical segment in each quadrant.