If a simple closed curve is composed of a finite number of circular arcs that are joined with continuous derivatives (C1 continuity), then it has an interesting property. Consider the face and hour-hand of a circular clock. Each point on the circular border of the clock face is uniquely related to a 12-hour time, and a normal vector pointing towards the center of the clock is in 1-1 correspondence with a 12-hour time. Now consider a "clock" whose border consists of a finite number of circular arcs joined in with C1 continuity and such that the center of the radius of curvature is always "to the right" as we traverse the curve in "clockwise" manner. Note that the radii of curvature of the different arc segments may be different. Yet the normal to any point on any circular arc which points towards the center of curvature is still related uniquely to the angle of the normal, and hence to a clock-time. So a point moving along such a border can always be given a unique clock time which is determined by its absolute angle relative to the center of curvature of its arc segment. Note that this is well-defined even at the join points, since both arcs agree on what time it is at their join points. Now it remains to be seen if we can construct such curves. We can obviously construct a circular arc with no join points. We can't construct one with one join point, unless it is still a single circle. We also can't construct one with two join points, unless the two radii of curvature match, because we can't achieve C1 continuity. I conjecture that such curves can always be constructed with 3 or more join points. The Colosseum has 8 such join points. Here is a way to get a "cheaper" Colosseum from 4 join points. We can construct two of the arcs to go from 2:00 o'clock to 4:00 o'clock and 8:00 o'clock to 10:00 o'clock. The 3rd and 4th arcs go from 4:00 o'clock to 8:00 o'clock and from 10:00 o'clock to 2:00 o'clock. We choose radius of curvature r1 for the first two arcs and radius of curvature for the second two arcs. If r1=r2, we get a circular clock face, but if r1<r2 or r1>r2, then we get an approximation to an elliptical clock face. Note that C0 continuity is assured by symmetry of the structure, while C1 continuity is assured by the continuity of the angles at the join points. Note that all the join points are determined by symmetry, once we pick on join "time" -- e.g., 2:00 o'clock. However, we are free to pick this particular join point anywhere between 12:00 o'clock and 3:00 o'clock. Using the same ideas, we can interpolate 4 additional join points & choose one additional radius of curvature r3. Once again, we can choose this third radius of curvature without further constraint. When we allow infinite radii of curvature, things get more interesting; also, if we allow negative radii of curvature, we can start "counting the clock backward" for those arc segments.