Documentation for line angles

docs/user_guide/ug_cable_model.md

@@ -35,6 +35,59 @@ $$
 
 Another way to see the cable plane is to rotate the cable plane.
 
+### Line angles and arm lengths
+
+![Image not available](./assets/angle_line_direct.drawio.png "Cable plane image")
+
+We want to compute $a_{cable}$ and $b_{cable}$ depending on $a$, the line angles, the arm lengths and the chain deplacements
+
+$A_i$ and $A_{i+1}$ (**attachments points**) are the points where cable is attached at the end of the suspension chains, respectively on left and right support. These are the points we want at the end.
+
+$E_i$ and $E_{i+1}$ (**edge_arm points**) are the points of the end of the arms, respectively on left and right support
+
+$C_i$ and $C_{i+1}$ (**center_arm points**) are the points on the support where the arms are attached
+
+$G_i$ and $G_{i+1}$ (**support ground points**) are the points on the ground where the support is set. More precisely, this point is the barycenter of the legs of the support.
+
+
+![Image not available](./assets/angle_line_points.drawio.png "Cable plane image up")
+
+![Image not available](./assets/angle_line_side.drawio.png "Cable plane image side")
+
+
+$a_{cable} = \sqrt{(x_{A_i}-x_{A_{i+1}})^2 + (y_{A_i}-y_{A_{i+1}})^2}$
+
+$b_{cable} = |z_{A_i}z_{A_{i+1}}|$
+
+
+We want the coordinates of $A_i$ and $A_{i+1}$ in the span frame of the support $i$
+
+
+
+#### Computing support ground $G_i$ coordinates
+
+Support ground coordinates $G_i$ are built recursively. Starting from $G_i$, we can build $\overrightarrow{G_{i}G_{i+1}}$, then get coordinates of $G_{i+1}$
+
+$\overrightarrow{G_{i}G_{i+1}}$ length is $a$. Its direction can be determined using the line angles $\gamma_i$
+
+Then center arm coordinates $C_i$ can be easily computed by adding the altitude of the arm.
+
+
+#### Computing edge arms $E_i$ coordinates
+
+In order to get $E_i$, we use point $E_i'$, which is the equivalent point if there wasn't any line angle.\
+Coordinates of $E_i'$ are $(x_{C_i}, L_l, z_{C_i})$. Same x and z coordinates than center arm points, but with an arm length along the y axis.\
+Then we can get $E_i$ by applying a rotation of $-\frac{\gamma_{l}}{2}$ around the z axis.
+
+Same thing for $E_{i+1}$, we use point $E_{i+1}'$.\
+Coordinates of $E_{i+1}'$ are $(x_{C_{i+1}}, L_l, z_{C_{i+1}})$.\
+Then we can get $E_{i+1}$ by applying a rotation of $+\frac{\gamma_{l}}{2}$ around the z axis.
+
+#### Computing attachement $A_i$ coordinates
+
+From coordinates of $E_i$ and $E_{i+1}$, you can get coordinates of $A_i$ and $A_{i+1}$, by adding $dep_x$ and $dep_y$ to their coordinates. 
+
+$dep_z$ can be calculated using the two other coordinates and the insulator length (TODO).
 
 ### Catenary model