多角形D上の点から任意の点への距離の二乗の平均値(グリーンの定理を用いて面積分を周回積分にして求める)

グリーンの定理(Green's theorem)

閉曲線$C$で囲まれた領域$D$を考える場合，$C^1$級凾数$P(x,y),Q(x,y)$について以下が成り立つ。 $$\begin{array}{rcl}{\iint }_D(\frac{\partial Q(x,y)}{\partial x}-\frac{\partial P(x,y)}{\partial y})\mathrm{d}x\mathrm{d}y& =& {\oint }_{C}P(x,y)\mathrm{d}x+Q(x,y)\mathrm{d}y\end{array}$$

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面積分を周回積分へ変形し，区間毎の線積分に展開する

点$x,y$から点$(p,q)$までの距離の二乗は $$\begin{array}{rcl}⟨r^2⟩& =& {\left\{\sqrt{{(x-p)}^2+{(y-q)}^2}\right\}}^2\\ & =& {(x-p)}^2+{(y-q)}^2\end{array}$$ であり，多角形D上の点($x,y$)から任意の点($p,q$)への距離の二乗の平均値$⟨r^2⟩$は， $$\begin{array}{rcl}⟨r^2⟩& =& {\iint }_D\rho (x,y)\{{(x-p)}^2+{(y-q)}^2\};\mathrm{d}x\mathrm{d}y\\ & =& {\iint }_D\frac{1}{A}{(x-p)}^2+{(y-q)}^2\mathrm{d}x\mathrm{d}y\cdots \mathrm{密}\mathrm{度}\mathrm{は}\mathrm{均}\mathrm{一}\mathrm{と}\mathrm{仮}\mathrm{定}:\rho (x,y)=\frac{1}{A}(\mathrm{定}\mathrm{数},A:D\mathrm{の}\mathrm{面}\mathrm{積})\\ & =& \frac{1}{A}{\iint }_D{(x-p)}^2+{(y-q)}^2\mathrm{d}x\mathrm{d}y\cdots {\int }_{X}cA(x)\mathrm{d}x=c{\int }_{X}A(x)\mathrm{d}x\end{array}$$ である．

これをグリーンの定理の式で満たすために例えば $$\begin{array}{rcl}Q(x,y)& =& \int {(x-p)}^2\mathrm{d}x\\ P(x,y)& =& -\int {(y-q)}^2\mathrm{d}y\end{array}$$ とおく．これは $$\begin{array}{rcl}\frac{\partial Q(x,y)}{\partial x}-\frac{\partial P(x,y)}{\partial y}& =& \int {(x-p)}^2\mathrm{d}x-\{-\int {(y-q)}^2\mathrm{d}y\}\\ & =& {(x-p)}^2+{(y-q)}^2\end{array}$$ となり，グリーンの定理の面積分側の被積分凾数を表現できている．
この$P,Q$を用いて線積分側の 被積分凾数を求めると以下のようになる． $$\begin{array}{rcl}⟨r^2⟩& =& {\int }_D\rho (x,y)r{(x,y;p,q)}^2\mathrm{d}x\mathrm{d}y\\ & =& {\int }_D\rho (x,y)\{{(x-p)}^2+{(y-q)}^2\}\mathrm{d}x\mathrm{d}y\\ & =& {\int }_D\frac{1}{A}\{{(x-p)}^2+{(y-q)}^2\}\mathrm{d}x\mathrm{d}y\\ & =& \frac{1}{A}{\int }_D\{{(x-p)}^2+{(y-q)}^2\}\mathrm{d}x\mathrm{d}y\\ & =& \frac{1}{A}\left[{\int }_D[\frac{\partial }{\partial x}\{\int {(x-p)}^2\mathrm{d}x\}-\frac{\partial }{\partial y}\{-\int {(y-q)}^2\mathrm{d}y\}]\mathrm{d}x\mathrm{d}y\right]\\ & =& \frac{1}{A}[{\oint }_C\{\int {(x-p)}^2\mathrm{d}x\}\mathrm{d}y+\{-\int {(y-q)}^2\mathrm{d}y\}\mathrm{d}x]\\ & =& \frac{1}{A}\{{\oint }_C\frac{1}{3}{(x-p)}^3\mathrm{d}y-\frac{1}{3}{(y-q)}^3\mathrm{d}x\}\\ & =& \frac{1}{A}\{{\oint }_C\frac{1}{3}{(x-p)}^3\mathrm{d}y-{\oint }_C\frac{1}{3}{(y-q)}^3\mathrm{d}x\}\\ & =& \frac{1}{A}\{\frac{1}{3}{\oint }_C{(x-p)}^3\mathrm{d}y-\frac{1}{3}{\oint }_C{(y-q)}^3\mathrm{d}x\}\end{array}$$

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$\left\{\right\}$内の第一項 区間毎の線積分

$$\begin{array}{rcl}\frac{1}{3}{\oint }_C{(x-p)}^3\mathrm{d}y& =& \frac{1}{3}\sum _{k=1}^n{\int }_{y_{k-1}}^{y_k}{\{({ϵ}_{k}y+{\zeta }_k)-p\}}^3\mathrm{d}y\cdots x={ϵ}_{k}y+{\zeta }_k\mathrm{多}\mathrm{角}\mathrm{形}\mathrm{の}\mathrm{区}\mathrm{間}(\mathrm{頂}\mathrm{点}k-1\mathrm{か}\mathrm{ら}k)\mathrm{毎}\mathrm{の}\mathrm{直}\mathrm{線}\mathrm{の}\mathrm{式}\\ & =& \frac{1}{3}\sum _{k=1}^n{\int }_{y_{k-1}}^{y_k}\{{({ϵ}_{k}y+{\zeta }_k)}^3-3{({ϵ}_{k}y+{\zeta }_k)}^{2}p+3({ϵ}_{k}y+{\zeta }_k)p^2-p^3\}\mathrm{d}y\cdots (A-B{)}^3=A^3-3A^{2}B+3A{B}^2-B^3\\ & =& \frac{1}{3}\sum _{k=1}^n{[\frac{1}{4{ϵ}_k}{({ϵ}_{k}y+{\zeta }_k)}^4-3\frac{1}{3{ϵ}_k}{({ϵ}_{k}y+{\zeta }_k)}^{3}p+3\frac{1}{2{ϵ}_k}{({ϵ}_{k}y+{\zeta }_k)}^2{p}^2-y{p}^3]}_{y_{k-1}}^{y_k}\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x,{\int }_a^b{x}^a\mathrm{d}x={\left[\frac{1}{a+1}{x}^{a+1}\right]}_a^b\\ & =& \frac{1}{3}\sum _{k=1}[\frac{1}{4{ϵ}_k}\{{({ϵ}_k{y}_k+{\zeta }_k)}^4-{({ϵ}_k{y}_{k-1}+{\zeta }_k)}^4\}-\frac{1}{{ϵ}_k}\{{({ϵ}_k{y}_k+{\zeta }_k)}^3-{({ϵ}_k{y}_{k-1}+{\zeta }_k)}^3\}p+\frac{3}{2{ϵ}_k}\{{({ϵ}_k{y}_k+{\zeta }_k)}^2-{({ϵ}_k{y}_{k-1}+{\zeta }_k)}^2\}{p}^2-(y_k-y_{k-1})p^3]\\ & =& \frac{1}{3}\sum _{k=1}^n[\frac{1}{4{ϵ}_k}\{{ϵ}_k^4(y_k^4-y_{k-1}^4)+4{ϵ}_k^3{\zeta }_k(y_k^3-y_{k-1}^3)+6{ϵ}_k^2{\zeta }_k^2(y_k^2-y_{k-1}^2)+4{ϵ}_k{\zeta }_k^3(y_k-y_{k-1})\}-\frac{1}{{ϵ}_k}\{{ϵ}_k^3(y_k^3-y_{k-1}^3)+3{ϵ}_k^2{\zeta }_k(y_k^2-y_{k-1}^2)+3{ϵ}_k{\zeta }_k^2(y_k-y_{k-1})\}p+\frac{3}{2{ϵ}_k}\{{ϵ}_k^2(y_k^2-y_{k-1}^2)+2{ϵ}_k{\zeta }_k(y_k-y_{k-1})\}{p}^2-(y_k-y_{k-1})p^3]\\ & & \cdots (ax+b{)}^4-(ay+b{)}^4=a^4{x}^4-a^4{y}^4+4a^{3}b{x}^3-4a^{3}b{y}^3+6a^2{b}^2{x}^2-6a^2{b}^2{y}^2+4a{b}^{3}x-4a{b}^{3}y=a^4(x^4-y^4)+4a^{3}b(x^3-y^3)+6a^2{b}^2(x^2-y^2)+4a{b}^3(x-y)\\ & & \cdots (ax+b{)}^3-(ay+b{)}^3=a^3{x}^3-a^3{y}^3+3a^{2}b{x}^2-3a^{2}b{y}^2+3a{b}^{2}x-3a{b}^{2}y=a^3(x^3-y^3)+3a^{2}b(x^2-y^2)+3a{b}^2(x-y)\\ & & \cdots (ax+b{)}^2-(ay+b{)}^2=a^2{x}^2-a^2{y}^2+2abx-2aby=a^2(x^2-y^2)+2ab(x-y)\\ & =& \frac{1}{3}\sum _{k=1}^n[\frac{1}{4{ϵ}_k}\{{ϵ}_k^4(y_k-y_{k-1})(y_k+y_{k-1})(y_k^2+y_{k-1}^2)+4{ϵ}_k^3{\zeta }_k(y_k-y_{k-1})(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+6{ϵ}_k^2{\zeta }_k^2(y_k-y_{k-1})(y_k+y_{k-1})+4{ϵ}_k{\zeta }_k^3(y_k-y_{k-1})\}-\frac{p}{{ϵ}_k}\{{ϵ}_k^3(y_k-y_{k-1})(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+3{ϵ}_k^2{\zeta }_k(y_k-y_{k-1})(y_k+y_{k-1})+3{ϵ}_k{\zeta }_k^2(y_k-y_{k-1})\}+\frac{3p^2}{2{ϵ}_k}\{{ϵ}_k^2(y_k-y_{k-1})(y_k+y_{k-1})+2{ϵ}_k{\zeta }_k(y_k-y_{k-1})\}-p^3(y_k-y_{k-1})]\\ & & \cdots (a^4-b^4)=(a-b)(a+b)(a^2+b^2)\\ & & \cdots (a^3-b^3)=(a-b)(a^2+ab+b^2)\\ & & \cdots (a^2-b^2)=(a-b)(a+b)\\ & =& \frac{1}{3}\sum _{k=1}^n(y_k-y_{k-1})[\frac{1}{4{ϵ}_k}\{{ϵ}_k^4(y_k+y_{k-1})(y_k^2+y_{k-1}^2)+4{ϵ}_k^3{\zeta }_k(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+6{ϵ}_k^2{\zeta }_k^2(y_k+y_{k-1})+4{ϵ}_k{\zeta }_k^3\}-\frac{p}{{ϵ}_k}\{{ϵ}_k^3(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+3{ϵ}_k^2{\zeta }_k(y_k+y_{k-1})+3{ϵ}_k{\zeta }_k^2\}+\frac{3p^2}{2{ϵ}_k}\{{ϵ}_k^2(y_k+y_{k-1})+2{ϵ}_k{\zeta }_k\}-p^3]\\ & =& \frac{1}{3}\sum _{k=1}^n(y_k-y_{k-1})[\frac{1}{4}\{{ϵ}_k^3(y_k+y_{k-1})(y_k^2+y_{k-1}^2)+4{ϵ}_k^2{\zeta }_k(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+6{ϵ}_k{\zeta }_k^2(y_k+y_{k-1})+4{\zeta }_k^3\}-p\{{ϵ}_k^2(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+3{ϵ}_k{\zeta }_k(y_k+y_{k-1})+3{\zeta }_k^2\}+\frac{3p^2}{2}\{{ϵ}_k(y_k+y_{k-1})+2{\zeta }_k\}-p^3]\end{array}$$

${ϵ}_k,{\zeta }_k$の代入と整理

$$\begin{array}{rcl}{ϵ}_k& =& \frac{x_k-x_{k-1}}{y_k-y_{k-1}}\\ {\zeta }_k& =& \frac{(x_{k-1}{y}_k-x_k{y}_{k-1})}{(y_k-y_{k-1})}\\ \\ & & {ϵ}_k^3(y_k+y_{k-1})(y_k^2+y_{k-1}^2)+4{ϵ}_k^2{\zeta }_k(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+6{ϵ}_k{\zeta }_k^2(y_k+y_{k-1})+4{\zeta }_k^3\\ & =& \frac{(x_k-x_{k-1}{)}^3}{(y_k-y_{k-1}{)}^3}(y_k+y_{k-1})(y_k^2+y_{k-1}^2)+4\frac{(x_k-x_{k-1}{)}^2}{(y_k-y_{k-1}{)}^2}\frac{(x_{k-1}{y}_k-x_k{y}_{k-1})}{(y_k-y_{k-1})}(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+6\frac{(x_k-x_{k-1})}{(y_k-y_{k-1})}\frac{(x_{k-1}{y}_k-x_k{y}_{k-1}{)}^2}{(y_k-y_{k-1}{)}^2}(y_k+y_{k-1})+4\frac{(x_{k-1}{y}_k-x_k{y}_{k-1}{)}^3}{(y_k-y_{k-1}{)}^3}\\ & =& \frac{1}{(y_k-y_{k-1}{)}^3}\{(x_k-x_{k-1}{)}^3(y_k+y_{k-1})(y_k^2+y_{k-1}^2)+4(x_k-x_{k-1}{)}^2(x_{k-1}{y}_k-x_k{y}_{k-1})(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+6(x_k-x_{k-1})(x_{k-1}{y}_k-x_k{y}_{k-1}{)}^2(y_k+y_{k-1})+4(x_{k-1}{y}_k-x_k{y}_{k-1}{)}^3\}\\ & =& (x_k+x_{k-1})(x_k^2+x_{k-1}^2)\\ \\ & & {ϵ}_k^2(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+3{ϵ}_k{\zeta }_k(y_k+y_{k-1})+3{\zeta }_k^2\\ & =& \frac{(x_k-x_{k-1}{)}^2}{(y_k-y_{k-1}{)}^2}(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+3\frac{(x_k-x_{k-1})}{(y_k-y_{k-1})}\frac{(x_{k-1}{y}_k-x_k{y}_{k-1})}{(y_k-y_{k-1})}(y_k+y_{k-1})+3\frac{(x_{k-1}{y}_k-x_k{y}_{k-1}{)}^2}{(y_k-y_{k-1}{)}^2}\\ & =& \frac{1}{(y_k-y_{k-1}{)}^2}\{(x_k-x_{k-1}{)}^2(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+3(x_k-x_{k-1})(x_{k-1}{y}_k-x_k{y}_{k-1})(y_k+y_{k-1})+3(x_{k-1}{y}_k-x_k{y}_{k-1}{)}^2\}\\ & =& x_k^2+x_k{x}_{k-1}+x_{k-1}^2\\ \\ & & {ϵ}_k(y_k+y_{k-1})+2{\zeta }_k\\ & =& \frac{(x_k-x_{k-1})}{(y_k-y_{k-1})}(y_k+y_{k-1})+2\frac{(x_{k-1}{y}_k-x_k{y}_{k-1})}{(y_k-y_{k-1})}\\ & =& \frac{1}{(y_k-y_{k-1})}\{(x_k-x_{k-1})(y_k+y_{k-1})+2(x_{k-1}{y}_k-x_k{y}_{k-1})\}\\ & =& x_k+x_{k-1}\end{array}$$

$\left\{\right\}$内の第一項　整理後

$$\begin{array}{rcl}\frac{1}{3}{\oint }_C{(x-p)}^3\mathrm{d}y& =& \frac{1}{3}\sum _{k=1}^n(y_k-y_{k-1})[\frac{1}{4}\{{ϵ}_k^3(y_k+y_{k-1})(y_k^2+y_{k-1}^2)+4{ϵ}_k^2{\zeta }_k(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+6{ϵ}_k{\zeta }_k^2(y_k+y_{k-1})+4{\zeta }_k^3\}-p\{{ϵ}_k^2(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+3{ϵ}_k{\zeta }_k(y_k+y_{k-1})+3{\zeta }_k^2\}+\frac{3p^2}{2}\{{ϵ}_k(y_k+y_{k-1})+2{\zeta }_k\}-p^3]\\ & =& \frac{1}{3}\sum _{k=1}^n(y_k-y_{k-1})[\frac{1}{4}\{(x_k+x_{k-1})(x_k^2+x_{k-1}^2)\}-p(x_k^2+x_k{x}_{k-1}+x_{k-1}^2)+\frac{3p^2}{2}(x_k+x_{k-1})-p^3]\\ & =& \frac{1}{12}\sum _{k=1}^n(y_k-y_{k-1})\{(x_k+x_{k-1})(x_k^2+x_{k-1}^2)\}-\frac{p}{3}\sum _{k=1}^n(y_k-y_{k-1})(x_k^2+x_k{x}_{k-1}+x_{k-1}^2)+\frac{p^2}{2}\sum _{k=1}^n(y_k-y_{k-1})(x_k+x_{k-1})-\frac{p^3}{3}\sum _{k=1}^n(y_k-y_{k-1})\\ & =& \frac{1}{12}\sum _{k=1}^n(y_k-y_{k-1})\{(x_k+x_{k-1})(x_k^2+x_{k-1}^2)\}-\frac{p}{3}\sum _{k=1}^n(y_k-y_{k-1})(x_k^2+x_k{x}_{k-1}+x_{k-1}^2)+\frac{p^2}{2}\sum _{k=1}^n(y_k-y_{k-1})(x_k+x_{k-1})\cdots \sum _{k=1}^n(y_k-y_{k-1})=0(\mathrm{周}\mathrm{回}\mathrm{積}\mathrm{分}\mathrm{な}\mathrm{の}\mathrm{で}y\mathrm{軸}\mathrm{の}\mathrm{移}\mathrm{動}\mathrm{量}\mathrm{の}\mathrm{和}\mathrm{は}\mathrm{一}\mathrm{周}\mathrm{し}\mathrm{た}\mathrm{ら}\mathrm{元}\mathrm{に}\mathrm{戻}\mathrm{る})\end{array}$$

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$\left\{\right\}$内の第二項　整理後

$$\begin{array}{rcl}-\frac{1}{3}{\oint }_C{(y-q)}^3\mathrm{d}x& =& -[\frac{1}{12}\sum _{k=1}^n(x_k-x_{k-1})\{(y_k+y_{k-1})(y_k^2+y_{k-1}^2)\}-\frac{q}{3}\sum _{k=1}^n(x_k-x_{k-1})(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+\frac{q^2}{2}\sum _{k=1}^n(x_k-x_{k-1})(y_k+y_{k-1})]\end{array}$$

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項毎に計算したものをまとめる

$$\begin{array}{rcl}⟨r^2⟩& =& \frac{1}{A}\{\frac{1}{3}{\oint }_C{(x-p)}^3\mathrm{d}y-\frac{1}{3}{\oint }_C{(y-q)}^3\mathrm{d}x\}\\ & =& \frac{1}{A}[[\frac{1}{12}\sum _{k=1}^n(y_k-y_{k-1})\{(x_k+x_{k-1})(x_k^2+x_{k-1}^2)\}-\frac{p}{3}\sum _{k=1}^n(y_k-y_{k-1})(x_k^2+x_k{x}_{k-1}+x_{k-1}^2)+\frac{p^2}{2}\sum _{k=1}^n(y_k-y_{k-1})(x_k+x_{k-1})]-[\frac{1}{12}\sum _{k=1}^n(x_k-x_{k-1})\{(y_k+y_{k-1})(y_k^2+y_{k-1}^2)\}-\frac{q}{3}\sum _{k=1}^n(x_k-x_{k-1})(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)+\frac{q^2}{2}\sum _{k=1}^n(x_k-x_{k-1})(y_k+y_{k-1})]]\\ & =& \frac{1}{A}\left[[\frac{1}{12}\sum _{k=1}^n(y_k-y_{k-1})\{(x_k+x_{k-1})(x_k^2+x_{k-1}^2)\}-\frac{1}{12}\sum _{k=1}^n(x_k-x_{k-1})\{(y_k+y_{k-1})(y_k^2+y_{k-1}^2)\}-2p\frac{1}{6}\sum _{k=1}^n(y_k-y_{k-1})(x_k^2+x_k{x}_{k-1}+x_{k-1}^2)-2q\{-\frac{1}{6}\sum _{k=1}^n(x_k-x_{k-1})(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)\}+p^2\frac{1}{2}\sum _{k=1}^n(y_k-y_{k-1})(x_k+x_{k-1})+q^2\{-\frac{1}{2}\sum _{k=1}^n(x_k-x_{k-1})(y_k+y_{k-1})\}]\right]\\ & =& \frac{1}{A}[\lambda +\mu -2pA{X}_G-2qA{Y}_G+p^{2}A+q^{2}A]\\ & & \cdots \lambda =\frac{1}{12}\sum _{k=1}^n(y_k-y_{k-1})\{(x_k+x_{k-1})(x_k^2+x_{k-1}^2)\}\\ & & \cdots \mu =-\frac{1}{12}\sum _{k=1}^n(x_k-x_{k-1})\{(y_k+y_{k-1})(y_k^2+y_{k-1}^2)\}\\ & & \cdots X_G=\frac{1}{6A}\sum _{k=1}^n(y_k-y_{k-1})(x_k^2+x_k{x}_{k-1}+x_{k-1}^2)(\mathrm{重}\mathrm{心}X\mathrm{軸}\mathrm{位}\mathrm{置})\\ & & \cdots Y_G=-\frac{1}{6A}\sum _{k=1}^n(x_k-x_{k-1})(y_k^2+y_k{y}_{k-1}+y_{k-1}^2)(\mathrm{重}\mathrm{心}Y\mathrm{軸}\mathrm{位}\mathrm{置})\\ & & \cdots A=\frac{1}{2}\sum _{k=1}^n(y_k-y_{k-1})(x_k+x_{k-1})(\mathrm{面}\mathrm{積})\\ & & \cdots A=-\frac{1}{2}\sum _{k=1}^n(x_k-x_{k-1})(y_k+y_{k-1})(\mathrm{面}\mathrm{積})\\ & =& \frac{\lambda +\mu }{A}-2p{X}_G-2q{Y}_G+p^2+q^2\\ & =& \frac{\lambda +\mu }{A}+X_G^2-2p{X}_G+p^2+Y_G^2-2q{Y}_G+q^2-X_G^2-Y_G^2\\ & =& \frac{\lambda +\mu }{A}+{(p-X_G)}^2+{(q-Y_G)}^2-X_G^2-Y_G^2\end{array}$$ よって$⟨r^2⟩$は多角形によって事前に決定される$A,X_G,Y_G,\lambda ,\mu$と，点$(p,q)$が与えられることで決まる．

$⟨r^2⟩$の最小値

$⟨r^2⟩$の最小値となる点$(u,v)$は，$(p-X_G),(p-X_G)$が0となる点$(X_G,Y_G)$である．

また，点$(X_G,Y_G)$から$⟨r^2⟩$の等しい点$(p,q)$は，以下の式を満たすことになる． $$\begin{array}{rcl}⟨r^2⟩& =& \frac{\lambda +\mu }{A}+{(p-X_G)}^2+{(q-Y_G)}^2-X_G^2-Y_G^2\\ {(p-X_G)}^2+{(q-Y_G)}^2& =& ⟨r^2⟩+X_G^2+Y_G^2-\frac{\lambda +\mu }{A}\\ \sqrt{{(p-X_G)}^2+{(q-Y_G)}^2}& =& \sqrt{⟨r^2⟩+X_G^2+Y_G^2-\frac{\lambda +\mu }{A}}\end{array}$$