When a circle rolls without slipping once around the outside of a polygon or a row of circles, its center traces a path with a shape different from the original figure. Roll the circle with the slider and see how its center moves differently at convex and concave corners.
Key principle — When the circle rolls in a straight line along an edge, its center also traces a line parallel to that edge, with almost the same length as the original edge (it is simply shifted outward by the radius).
When the circle reaches a convex vertex (where the interior angle is less than 180°), its center changes direction by tracing an arc of radius r centered at that vertex. The central angle of this arc is exactly the vertex’s exterior angle (180°−interior angle).
Conversely, at a concave vertex (where the interior angle is greater than 180°, a reflex angle), the circle does not rotate at all and passes smoothly from one edge to the next. Therefore, for shapes with concave parts, both the number of arcs and the length of the straight sections differ from those of the original shape when calculating the locus length — check it directly below.
Choose a shape. For convex polygons such as an equilateral triangle or rectangle, the sum of the arc central angles is always 360°. For shapes with concave parts, such as the arrow and L-shape, the sum is greater than 360°, and no arc is formed at a concave vertex.
Place several equal-sized circles in a row and roll another circle around the outside of them. Change the number of circles (N) and the two radii to see how the center’s path changes. At every point where two circles touch, a concave gap (groove) forms. How deeply the rolling circle dips into the gap is determined by the ratio of the two radii — increase the fixed-circle radius (R) and decrease the rolling-circle radius (r) to see it dip much deeper.