What we furthermore need is the following Lemma.
wa(x−A)2+wb(x−B)2=(wa+wb)(x−waA+wbBwa+wb)2+wawbwa+wb(A−B)2
Where x,A,B and wa,wb>0 are real numbers.
Proof.
wa(x−A)2+wb(x−B)2=wa(x2−2Ax+A2)+wb(x2−2Bx+B2)=(wa+wb)x2−2(waA+wbB)x+(waA2+wbB2)=(wa+wb)[x2−2waA+wbBwa+wb+(waA+wbBwa+wb)2]−(wa+wb)(waA+wbBwa+wb)2+(waA2+wbB2)=(wa+wb)(x−waA+wbBwa+wb)2+−(waA+wbB)2+(wa+wb)(waA2+wbB2)wa+wb=(wa+wb)(x−waA+wbBwa+wb)2+−(w2aA2+2wawbAB+w2bB2)+w2aA2+w2bB2+wawbA2+wawbB2wa+wb=(wa+wb)(x−waA+wbBwa+wb)2+wawb(A2−2AB+B2)wa+wb=(wa+wb)(x−waA+wbBwa+wb)2+wawbwa+wb(A−B)2
Without Loss Of Generality (WLOG) we can put for the weighting factors: w1+w2=1 . Then we have:
wa(x−A)2+wb(x−B)2=[x−(waA+wbB)]2+wawb(A−B)2
In particular when w1=w2=1/2 :
12[(x−A)2+(x−B)2]=(x−A+B2)2+(A−B2)2
Application. We have found for the edges:
Ri×[2.sin(π/M),(c−1),c.2.sin(π/M),(c−1)]
What we want now is to determine the constant c in such a way that the following approximate
equalities hold as good as possible:
2.sin(π/M)≈c−1andc.2.sin(π/M)≈c−1
Meaning that the quadrilaterals are similar to squares as good as possible.
Formulated in a Least Squares sense, this in turn means that:
12[(x−A)2+(x−B)2]=minimum(c)
with the substitutions x=c−1 , A=2.sin(π/M) and B=c.2.sin(π/M) .
The minimum is obviously attained when x=(A+B)/2 or:
c−1=sin(π/M)+csin(π/M)⟹c[1−sin(π/M)]=1+sin(π/M)⟹1+sin(π/M)1−sin(π/M)=c=(RNR0)1/N⟹RNR0=(1+sin(π/M)1−sin(π/M))N
If the outer mesh radius is normed to one, ipse est RN=1,
then we have for the inner mesh radius:
R0=(1−sin(π/M)1+sin(π/M))N