# Modelling of Evenness of Runways as an Element of Sustainable Airport Maintenance
## 2 Methods for Measuring the Evenness of Airport Runways
### Methods for Modelling the Evenness of Airport Runways
#### 2.1.1 Approximation with Regression Plane  
The calculation of the plane equation is possible if at least three three-dimensional points of the plane exist. The approximation of the plane tries to find the equation of the plane that best matches the set of points (the point cloud). The plane equation was determined for non-planar points using a Moore-Penrose matrix inverse least squares solution. Equations (1)-(3) show the general form of the equation of a plane with a normal vector (\(\overrightarrow{\mathbf{n}}\)) [55]:  
\[\mathbf{a}\cdot\mathbf{x}+\mathbf{b}\cdot\mathbf{y}+\mathbf{c}\cdot\mathbf{H} +\mathbf{D}=0 \tag{1}\]  
where:  
D--the distance between the plain and centroid of the cloud of point,  
a, b, c--the parameters in the equation of the plain  
x-x coordinate of the point  
y-y coordinate of the point  
H-normal orthometric altitude or altitude  
It is assumed that c = 1 (weight) and the equation of the plane is written in the matrix shape, where x is the vector of the plane parameters [55]:  
\[\mathbf{A}\cdot\mathbf{x}=\mathbf{B}\left[\begin{array}{ccc}\mathbf{x}_{0} &\mathbf{y}_{0}&1\\ \mathbf{x}_{1}&\mathbf{y}_{1}&1\\ \mathbf{x}_{0}&\mathbf{y}_{0}&1\end{array}\right]\cdot\left[\begin{array}{ c}\mathbf{a}\\ \mathbf{b}\\ \mathbf{D}\end{array}\right]=\left[\begin{array}{c}-\mathbf{H}_{0}\\ -\mathbf{H}_{1}\\ -\mathbf{H}_{n}\end{array}\right] \tag{2}\]  
Since we are looking for the equation of the plane through more than three points, the system is predefined. The solution is found using the left pseudoinverse matrix or Moore-Penrose matrix inverse A\({}^{+}\)[56], where N is the number of points:  
\[\mathbf{A}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}=\mathbf{A}^{ \mathrm{T}}\cdot\mathbf{B}\left[\begin{array}{ccc}\sum\mathbf{x}_{i}\cdot \mathbf{x}_{i}&\sum\mathbf{x}_{i}\cdot\mathbf{y}_{i}&\sum\mathbf{x}_{i}\\ \sum\mathbf{y}_{i}\cdot\mathbf{x}_{i}&\sum  
If it is assumed that \(\sum\ x_{i}=\sum\ y_{i}=\sum\ H_{i}=0\), we can conclude that D = 0 and the plane passes through the centroid of point cloud. The accuracy of the calculated parameters is given with the diagonal part in the variant-covariant matrix [56]:  
\[\sum=\left[\begin{array}{cc}\sigma_{a}^{2}&\sigma_{ab}\\ \sigma_{ba}&\sigma_{b}^{2}\end{array}\right] \tag{4}\]  
\(\sigma_{a}\)--the accuracy of determining the unknown a in the plane equation  
\(\sigma_{b}\)--the accuracy of determining the unknown b in the plane equation  
\(\sigma_{c}\)--the accuracy of determining the unknown c in the plane equation (\(\sigma_{c}=1\))  
\(\sigma_{D}\)--the accuracy of determining the unknown D in the plane equation (\(\sigma_{D}=1\))  
The calculated and presented regression plane represents a continuous nonlinear plane for which the squares of the perpendicular distances from all measured points to the regression plane are considered to be the smallest. The regression plane is defined on the entire interval \(X\) (\(-\infty<x\ <\infty\) ) in Y ( \(-\infty<y\ <\infty\)): however, its useful value is limited to the field of research.