A plot of the ratio between the actual period of a pendulum and the approximate value obtained for small angles, as a function of the amplitude. According to Pendulum (mathematics), the oscillation period for small angles is given by:

${\displaystyle T_{0}=2\pi {\sqrt {\ell \over g}}}$

while the actual period for any angle is given by:

${\displaystyle T=4{\sqrt {\ell \over g}}F\left({\sin {\frac {\theta _{0}}{2}}},{\pi \over 2}\right)}$

where:

${\displaystyle F(k,\phi )=\int _{0}^{\phi }{1 \over {\sqrt {1-k^{2}\sin ^{2}{\theta }}}}\,d\theta .}$

so the ratio is given by:

${\displaystyle {\frac {T}{T_{0}}}={\frac {2}{\pi }}F\left({\sin {\frac {\theta _{0}}{2}}},{\pi \over 2}\right)}$

and this is the function plotted in the graph. First, with the following Matlab code I created a file called pendulum_period.dat;

then, in order to plot it, I used the Gnuplot code.