In geometry, the midpoint-stretching polygon of a cyclic polygon is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the circular arcs between the vertices of .[1] It may be derived from the midpoint polygon of (the polygon whose vertices are the edge midpoints) by placing the polygon in such a way that the circle's center coincides with the origin, and stretching or normalizing the vector representing each vertex of the midpoint polygon to make it have unit length.
The midpoint-stretching polygon is also called the shadow of ; when the circle is used to describe a repetitive time sequence and the polygon vertices on it represent the onsets of a drum beat, the shadow represents the set of times when the drummer's hands are highest, and has greater rhythmic evenness than the original rhythm.[2]
The midpoint-stretching polygon of a regular polygon is itself regular, and iterating the midpoint-stretching operation on an arbitrary initial polygon results in a sequence of polygons whose shape converges to that of a regular polygon.[1]