# Analysis of Lightweight Structure Mesh Topology of Geodesic Domes
## 2 Shaping Geodesic Dome Mesh
### The First Method of Dividing a Regular Octahedron
#### 2.1.1 Determining the Nodal Points of a Dome Shaped according to the First Method  
First, the initial triangle of the regular octahedron was described on the first quarter of the hemisphere in the polar coordinate system. The range of analyzed meridian angles is \(\phi\in\) <0;90\({}^{\circ}\)>, as well as \(\lambda\in\) <0;90\({}^{\circ}\)> for the latitudinal angles. Taking into account the assumptions of the first method, the division into meridian angles was made by dividing the edges into equal sections, determining the angle between subsequent points using the following formula:  
\[\phi^{\prime}=\frac{90}{n+1}=\frac{90}{24+1}=3.75\ [^{\circ}], \tag{3}\]  
where:  
\(n\) is the number of the division points of the initial triangular octahedron face.  
Then, the formula for the angular ordinate of subsequent layers of nodal points was determined:  
\[\phi_{n}=(n_{w}-1)\cdot\phi^{\prime}\ \ [^{\circ}], \tag{4}\]  
where:  
\(n_{w}\) is the number of point layers, \(n_{w}\)\(\in\)\(N_{+}\).  
The obtained values of meridional angular ordinates are presented in Table 1 and Figure 3.  
\begin{table}
\begin{tabular}{c c} \hline \hline \(n_{w}\) [\(-\)] & \(\phi_{n}\) [\({}^{\circ}\)] \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 1: Values of meridional angular ordinates.  
In the case of latitudinal coordinates \(\lambda\), based on the analysis of Figure 2, the formula for the angle between subsequent points on the same layer was derived:  
\[\lambda^{\prime}_{w}=\frac{90}{n-(n_{w}-1)}\left[{}^{\circ}\right] \tag{5}\]  
Then, a general formula was determined for the latitudinal angular ordinate of the point for individual layers.  
\[\lambda^{k}_{n_{w}}=\{(n-n_{w}+k)\}\cdot\lambda^{\prime}_{w}-90\left[{}^{\circ}\right] \tag{6}\]  
where:  
\(k\) is the number of points in the layer \(k\in\langle 1;n+2-n_{w}\rangle\).  
According to the above formula, the latitudinal angular coordinate was determined for all dome layers except for the top, where there is only one point with Cartesian coordinates P(0;0;R), as shown in Figure 1.  
Table 2 presents the results for the exemplary first and fifth layers (Figure 3).  
The angular ordinates obtained in this way were grouped in pairs to create the angular coordinates of each node in one quarter of the analyzed dome (Figure 3). Therefore, each point received three factors describing its position in space according to the scheme \(P^{k}_{n_{w}}\)\((R,\phi,\lambda)\). For example, the location of the point described as fifth on layer five was given the format \(P^{5}_{5}\) (25.00 m; 15.00\({}^{\circ}\);18.00\({}^{\circ}\)).  
The coordinates obtained in this way create a cloud of points describing the nodes of the dome. To enter them into the numerical program, each point was converted to a Cartesian coordinate system. To move from the spherical coordinate system \(P^{k}_{n_{w}}\)\((R,\phi,\lambda)\) to the Cartesian coordinate system \(P(x\),\(y\),\(z)\), the transformation formulas were used:  
\[\begin{array}{l}x=R\cdot\sin\phi\cdot\cos\lambda,\\ y=R\cdot\sin\phi\cdot\sin\lambda,\\ z=R\cdot\sin\phi,\end{array} \tag{7}\]  
In the case of point \(P^{5}_{5}\) (25.00 m; 15.00\({}^{\circ}\);18.00\({}^{\circ}\)) using the above transformation rules, a point with coordinates in the Cartesian system was obtained:  
\[\begin{array}{l}x=25.00\cdot\sin 18^{\circ}\cdot\cos 15^{\circ}=7.46\ [\mathrm{m} ],\\ y=25.00\cdot\sin 18^{\circ}\cdot\sin 15^{\circ}=2.00\ [\mathrm{m}],\\ z=25.00\cdot\sin 18^{\circ}=7.73\ [\mathrm{m}],\end{array} \tag{8}\]  
The above procedure was applied to all points, making it possible to generate a mesh of the computational model in the numerical program.  
Figure 3: Meridian angular coordinates for the first 9 layers of points.