## 2 Related Works
### Point Cloud Representation
#### 2.2.4 The TLS Point Cloud in Cartographic Projection  
Cartographic transformations are also applied for processing the point cloud to the raster form. In mathematical cartography, the reference surface is assumed to be an ellipsoid or a sphere, as the so-called surface of the original. The term \"cartographic projection\" is understood as a regular transformation of a flattened rotatable ellipsoid or the transformation of a sphere into a plane [42]. In this paper, only the mathematical notation for projection of a sphere onto a plane is used, which may be described by the following Equations (3) and (4).  
Vector equation of a sphere:  
\[\begin{gathered}\overset{\rightarrow}{\mathrm{r}}=[X=S\cos \theta\cos\alpha,\ Y=S\cos\theta\sin\alpha,\ Z=S\sin\alpha]\\ \theta e\langle-\frac{\pi}{2},\frac{\pi}{2}\rangle;\ a\epsilon<- \pi,\pi)\end{gathered} \tag{3}\]  
The equation of an image of a sphere in a plane:  
\[\overset{\rightarrow}{\mathrm{r}}=[x=x(\theta,\alpha),\ y=y(\theta,\alpha)] \tag{4}\]  
where \(X,\ Y,\ Z\) are orthogonal coordinates on the sphere surface, \(\theta,\alpha,S\) are spherical coordinates on the sphere surface, and \(x,y\) are orthogonal coordinates on a plane.  
Projection regularity condition:  
For each point of a sphere the system of inequalities (5) must also be satisfied \[\begin{array}{c}\left|\stackrel{{\rightarrow}}{{r}}_{0}\stackrel{{ \rightarrow}}{{r}}_{\alpha}\right|\
eq 0\\ \left|\stackrel{{\rightarrow}}{{r}}_{0}\stackrel{{ \rightarrow}}{{r}}_{\alpha}\right|\
eq 0\end{array} \tag{5}\]  
The cylindrical projection relies upon projecting a sphere onto the side of a cylinder, the rotation axis of which corresponds to the straight line, which connects the geographical poles.  
Transformation of a sphere onto a plane is expressed by Equation (6).  
\[\stackrel{{\rightarrow}}{{r}}=[\mathrm{x}=\mathrm{x}(\mathrm{B }),\ \mathrm{y}=\mathrm{c}\cdot\mathrm{L}] \tag{6}\]  
where c is the constant of the cylindrical projection, B is the geographical longitude, and L is the geographical latitude.  
In the cylindrical projections, images of parallels, B = constant, are presented in the form of straight lines parallel to the y-axis of the orthogonal coordinate system. Meridians, L = constant, are projected in the form of straight lines parallel to the x-axis of the coordinate system.  
During the terrestrial laser scanning data orientation process, raster images in Mercator cylindrical equal-angle projection were also applied. Conversion of point coordinates to the form of this projection is performed using Equation (7).  
\[\mathrm{x}=\mathrm{R}\ln\left[\tan\Bigl{(}\frac{\pi}{4}+\frac{\alpha}{2} \Bigr{)}\right] \tag{7}\]  
\[\mathrm{y}=\mathrm{R}\beta\]  
where R is the radius of a sphere (equal to one unit) and \(\alpha\), \(\beta\) are horizontal and vertical angles, respectively.  
In this projection, the grid is an orthogonal network. Images of meridians and parallels are straight lines parallel to respective axes of the planar coordinate system. Poles of the geographic system are projected to infinity. Therefore, this projection is used only within parallel zones, without circumpolar areas [42].  
Other cartographic projections have been analysed in detail by Reference [18].