I am researching some algorithms and it turns out that the following figure I made can model what is happening in a "step". I am not a mathematician, so I was having a hard time with this one. The question is, based on the figure below, if $\theta_{i,L}$, $\, \theta_{i,R}$, and $\, L_i$ are given, how can one calculate $A_i$, and then further $A_{i+1, L}$ and $A_{i+1, R}$? Where these are the heights of the 3 triangles in the picture below.
If it helps, at one point months ago I seemed to have calculated the following equations, but I don't remember how I found them and I haven't been able to recreate what I did to come up with them, so they could be garbage.
$ A_i = L_i \sin \theta_{i,R} \sin \theta_{i,L} \csc(\theta_{i,R} + \theta_{i,L} ) $
$ A_{i+1, L} = L_i \sin(\theta_{i,L}) \sin(\alpha_i \theta_{i,L}) \sin(\alpha_i \theta_{i,R}) \csc(\theta_{i,L} + \theta_{i,R}) \csc(\alpha_i (\theta_{i,L} + \theta_{i,R})) $
$ A_{i+1, R} = L_i \sin(\theta_{i,R}) \sin(\alpha_i \theta_{i,L}) \sin(\alpha_i \theta_{i,R}) \csc(\theta_{i,L} + \theta_{i,R}) \csc(\alpha_i (\theta_{i,L} + \theta_{i,R})) $
Where $\alpha$ is the fraction of the previous angle that the subsequent angle has, e.g. $ \theta_{i+1,L} = \alpha \theta_{i, L}$ where $0 \leq \alpha \leq 1$.
Additionally, it may or may not be relevant that $0 \leq \theta_{i,R}, \, \theta_{i,L} < 45^{\circ}$.
If I understood properly your problem, you shall start by imposing $$ \left\{ \matrix{ L_{\,i + 1,\;R} \cos \theta _{\,i,\;R} + L_{\,i + 1,\;L} \cos \theta _{\,i,\;L} = L_{\,i} \hfill \cr L_{\,i + 1,\;R} \sin \theta _{\,i,\;R} = L_{\,i + 1,\;L} \sin \theta _{\,i,\;L} = A_{\,i} \hfill \cr} \right. $$ which is a system of three equations in six unknowns.
Taking $L_i, \, \theta_{\, i, \, R}, \, \theta_{\, i, \, L}$ as given, then you can put the system into the matricial form as $$ \left( {\matrix{ {\cos \theta _{\,i,\;R} } & {\cos \theta _{\,i,\;L} } & 0 \cr {\sin \theta _{\,i,\;R} } & { - \sin \theta _{\,i,\;L} } & 0 \cr {\sin \theta _{\,i,\;R} } & 0 & { - 1} \cr } } \right)\left( {\matrix{ {L_{\,i + 1,\;R} } \cr {L_{\,i + 1,\;L} } \cr {A_{\,i} } \cr } } \right) = \left( {\matrix{ {L_{\,i} } \cr 0 \cr 0 \cr } } \right) $$ and solve it for $A_i, \, L_{\, i+1, \, R}, \, L_{\, i+1, \, L}$.
Then the smaller ones will follow the same track, with the angles changed by $\alpha$.