Sines and Cosines Over Long Periods of Time in Spacetime

summary:

Over long periods of time with smaller changes in space, sine and cosine functions make spirals.

description:

In classical physics, the amount of change in time measured in the same units as space vastly exceeds changes in space. In other words, relativistic velocities are low. In these animations, time changes by 100 while changes in space are only 20.

command:

q_graph -loop 0 -box 10 -dir vp -out sin-cos-50t -command 'q_add_n -50 -10 -10 -10 .05 .01 .01 .01 2000' -color yellow -command 'q_add_n -50 -10 -10 -10 .005 .001 .001 .001 20000 | q_sin' -color red -command 'q_add_n -50 -10 -10 -10 .005 .001 .001 .001 20000 | q_cos' -color blue

math

equation:

$\begin{align*} \sin(t,\vec{R}) &= (\sin(t) \cosh(|R|), \cos(t) \sinh(|R|) \frac{\vec{R}}{|R|})\\ \cos(t,\vec{R}) &= (\cos(t) \cosh(|R|), \sin(t) \sinh(|R|) \frac{\vec{R}}{|R|}) \end{align*}$

sage notebook:

Quaternion Trig Functions

numbers:

sin-cos-50t.povray.100.1000.numbers