Successive Averages of the Vertices of a Polygon
Successive Averages of the Vertices of a Polygon
If the vertices (thought of as vectors) of a polygon are , then the midpoints of the sides are . These new points form the vertices of another polygon, and the midpoints of its sides are . This process can be repeated for iterations.
{a,b,c,d,e,f,…}
{(a+b)/2,(b+c)/2,(c+d)/2,(d+e)/2,(e+f)/2,…}
{(a+2b+c)/2,(b+2c+d)/2,(c+2d+e)/2,…}
n
There is another variable, , the number of successive points around the polygon that can be averaged. Here are the next two initial averages (and ):
k
k=3
k=4
{(a+b+c)/3,(b+c+d)/3,(c+d+e)/3,…}
{(a+b+c+d)/4,(b+c+d+e)/4,(c+d+e+f)/4,…}
The original seven points are scattered randomly. For clarity, drag them to form a convex or star-shaped polygon and experiment with the controls.