## Appendix A
\begin{array}{l}y_{\max}=\langle y\rangle+A,\,y_{\min}=\langle y\rangle-A,\, \to A=\frac{1}{2}(y_{\max}-y_{\min}),\\ x_{\max}=\langle x\rangle+C,\,x_{\min}=\langle x\rangle-C,\,\to C=\frac{1}{2}( x_{\max}-x_{\min}).\end{array} \] (A10)  
Parameter \(B\) is found from relationships in Equations (A8) and (A9) as a positive root of the quadratic equation written as follows. \[ \begin{array}{l}B^{2}-2\big{\langle}\Delta x\Delta y\big{\rangle}B-\big{[}A ^{2}C^{2}-\big{\langle}\Delta x^{2}\big{\rangle}A^{2}-\big{\langle}\Delta y^{ 2}\big{\rangle}C^{2}\big{]}=0,\\ B=\big{\langle}\Delta x\Delta y\big{\rangle}+\Big{[}(\langle\Delta x\Delta y \rangle)^{2}+A^{2}C^{2}-\big{\langle}\Delta x^{2}\big{\rangle}A^{2}-\big{ \langle}\Delta y^{2}\big{\rangle}C^{2}\big{]}^{1/2}.\end{array} \] (A11)  
This root is chosen from the comparison of two identity sequences (\(x_{k}=y_{k}\)) that follows from the obvious requirement \(B=A^{2}\), (\(\alpha=0\)). Concluding this section, one can say that, with the help of the rotated counterclockwise ellipse in Equation (A7), we reduced \(2n\) random points figuring in equation (A4) to eight statistical parameters \(\text{Pr}_{8}\)(2): (\(<\)\(x^{p}\)\(>\), \(<\)\(y^{p}\)\(>\), \(p=1\),\(2\); \(<\)\(xy\)\(>\), \(A\), \(C\), \(\alpha\)). Equations (A7)-(A9) can be considered as the geometrical interpretation of the conventional Pearson correlation coefficient defined by Equation (A9). In order to increase the sensitivity in the detection of the hidden distortions that are contained in two random curves, one can increase the order of the calculated moments. For this purpose, one can consider the invariant of the fourth order. \[L_{k}^{(4)}=A_{4}(x-x_{k})^{4}+2B_{4}(x-x_{k})^{2}(y-y_{k})^{2}+C_{4}(y-y_{k})^ {4}.\] (A12)  
Inserting Equation (A12) into Equation (A13),  
\[\frac{1}{n}\sum_{k=1}^{n}L_{k}^{(4)}=\,Inv,\] (A13)  
and equating the linear terms,  
\[X\equiv x-\langle x\rangle,\,Y\equiv y-\langle y\rangle,\] (A14)to zero, one can obtain the following combination:  
\[ \begin{array}{l}K(X,Y)=K_{2}(X,Y)+K_{4}(X,Y)=I_{4}.\\ K_{2}(X,Y)=A_{2}X^{2}-2B_{2}X\cdot Y+C_{2}Y^{2},\\ K_{4}(X,Y)=A_{4}X^{4}-2B_{4}X^{2}Y^{2}+C_{4}Y^{4},\\ I_{4}=A_{4}\Big{<}\big{(}(\Delta x)^{4}\big{>}-2B_{4}\Big{<}(\Delta x)^{2}( \Delta y)^{2}\Big{>}+C_{4}\Big{<}(\Delta y)^{4}\Big{>}.\end{array}  \tag{101}\]  
The combinations shown below define the constants figuring in the DGI (Equation (101)). \[ \begin{array}{l}B_{4}=\frac{\big{<}(\Delta x)^{3}\big{>}}{\big{<}(\Delta x) \big{>}^{2}}A_{4}\equiv\sigma_{B}A_{4},\\ C_{4}=\frac{\big{<}(\Delta x)^{3}\big{>}}{\big{<}(\Delta y)^{3}\big{>}}\frac{ \big{<}(\Delta y)(\Delta x)^{2}\big{>}}{\big{<}(\Delta x)(\Delta y)^{2}\big{>} }A_{4}\equiv\sigma_{C}A_{4},\\ A_{2}=\big{[}6\big{<}(\Delta x)^{2}\big{>}-2\sigma_{B}\Big{<}(\Delta y)^{2} \Big{>}\Big{]}A_{4}\equiv S_{A}A_{4},\\ B_{2}=4\sigma_{B}\big{<}\Delta x\Delta y\big{>}A_{4}\equiv S_{B}A_{4},\\ C_{2}=\big{[}6\sigma_{C}\Big{<}\big{<}(\Delta y)^{2}\big{>}-2\sigma_{B}\Big{<} (\Delta x)^{2}\Big{>}\Big{]}A_{4}\equiv S_{C}A_{4}.\end{array}  \tag{102}\]  
The constant \(I_{4}\) and \(Inv\) from Equation (100) are defined by Equation (101)  
\[Inv=2I_{4}=2\Big{[}\Big{<}\big{(}(\Delta x)^{4}\big{>}-2\sigma_{B}\Big{<}( \Delta x)^{2}(\Delta y)^{2}\Big{>}+\sigma_{C}\Big{<}(\Delta y)^{4}\Big{>} \Big{]}A_{4}. \tag{103}\]  
The averaged values \(<\! (\Delta x)^{q}(\Delta y)^{p}\!\!>\) characterizing two compared sets are defined by Equation (100). The polynomial \(K(X,Y)\) of the fourth order can be separated in the polar coordinate system. Using the notations in Equation (102) and taking into account the fact that the constant \(A_{4}\) figuring in \(K(X,Y)\) is an arbitrary proportion multiplier and, therefore, can be omitted, we present the desired curve in the form  
\[ \begin{array}{l}x(\varphi)=\langle x\rangle+r(\varphi)\cos\varphi,\\ y(\varphi)=\langle y\rangle+r(\varphi)\sin\varphi,\\ r(\varphi)=\bigg{[}\frac{\sqrt{q_{2}^{2}(\varphi)+4I_{4}q_{4}(\varphi)}-q_{2}( \varphi)}{2q_{4}(\varphi)}\bigg{]}^{1/2}.\end{array}