\[\begin{equation} \vec{p_{4n-3}}= \vec{p_{4n-4}}+ \left[ \begin{array}{cc} h_1 \cos{\beta_{2n-1}}\\ h_1 \sin{\beta_{2n-1}} \end{array} \right] \end{equation}\]
\[\begin{equation} \vec{p_{4n-2}}= \vec{p_{4n-3}}+ \left[ \begin{array}{cc} L \cos{\alpha_{2n-1}}\\ L \sin{\alpha_{2n-1}} \end{array} \right] \end{equation}\]
\[\begin{equation} \vec{p_{4n-1}}= \vec{p_{4n-2}}+ \left[ \begin{array}{cc} h_2 \cos{\beta_{2n}}\\ h_2 \sin{\beta_{2n}} \end{array} \right] \end{equation}\]
\[\begin{equation} \vec{p_{4n}}= \vec{p_{4n-1}}+ \left[ \begin{array}{cc} L \cos{\alpha_{2n}}\\ L \sin{\alpha_{2n}} \end{array} \right] \end{equation}\]
For angles \(\alpha_n\) and \(\beta_n\), following relationships can be established:
For n is odd,
\(\begin{equation} \alpha_{n+1}-\alpha_n = \{ \begin{array}{cc} 2 \theta_n + \pi,\; for \; \alpha_n<0 \;and\; \alpha_{n+1}>0\\ 2 \theta_n - \pi,\; else\\ \end{array} \end{equation}\)
\(\begin{equation} \beta_{n+1}= \{ \begin{array}{cc} -\frac{\pi}{2} + \theta_n + \alpha_n + 2 \pi,\; for \; \alpha_n<0 \;and\; \beta_{n+1}>0\\ -\frac{\pi}{2} + \theta_n + \alpha_n,\; else\\ \end{array} \end{equation}\)
For n is even,