Sine and Cosine for a Fixed Point in Space is a Circle

summary:

When a position in space is fixed, the sine and cosine functions circle that point, the angle depending on the exact values of x, y, and z.

description:

Sines and cosines have to do with circles. By fixing x, y, and z, the circle stays fixed. What direction the line in space points to is arbitrary.

The line in yellow is the input for the The length of the line in space is the amplitude.

command:
q_graph -loop 0 -box 25 -dir vp -out sin-cos_xyz_constant -command 'q_add_n -50 1 2 1 .05 0 0 0 2000' -color yellow -command 'q_add_n -50 1 2 1 .05 0 0 0 2000 | q_sin' -color red -command 'q_add_n -50 2 1 2 .01 0 0 0 10000 | q_cos | q_x_scalar 2' -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*}

tags
Physics Tag:
simple harmonic oscillator
Math Tag:
trig functions
sine
cosine
Programming Tag:
command line quaternions
q_sin
q_cos

Sine and Cosine from -5 to 5

summary:

Quaternions ranging from (-5, -5, -5, -5) to (5, 5, 5, 5) were fed into a sine and cosine function, with these odd looking results.

description:

The linear input is in yellow, the odd sine function is in red, the even cosine in blue. The cosine has only one apparent line diving into the origin because that line is a doublet. I thought I knew what cosines and sines looked like, but it feels like there is a frightening large amount of unknown diversity possible for these formerly familiar friends. The cause of the oddness maybe due to varying t, x, y, and z at the same time.

command:
q_graph -loop 0 -box 10 -dir vp -out sin-cos -command 'q_add_n -10 -10 -10 -10 .01 .01 .01 .01 2000' -color yellow -command 'q_add_n -10 -10 -10 -10 .01 .01 .01 .01 2000 | q_sin' -color red -command 'q_add_n -10 -10 -10 -10 .01 .01 .01 .01 2000 | 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*}

numbers:
tags
Physics Tag:
simple harmonic oscillator
Math Tag:
trig functions
sine
cosine
Programming Tag:
command line quaternions
q_sin
q_cos

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*}

tags
Physics Tag:
simple harmonic oscillator
Math Tag:
trig functions
sine
cosine
Programming Tag:
command line quaternions
q_sin
q_cos

Sines in Many Directions

summary:

When the input never moves in space, the output of oscillating points is easier to understand.

description:

Each of these sets of events starts out pointing in a different direction. Yet the x, y, and z values of the input is never altered. This is why you can spot the spatial origin, the point in the center of all the moving points. It would be simple enough to shift these arbitrary oscillators around precisely the origin to arbitrary oscillators around arbitrary points - just add in an arbitrary value as a last step.

command:
q_graph -loop 0 -box 12 -dir vp -out sines_xyz_constant -command 'q_add_n -50 2 2 1 .05 0 0 0 2000 | q_sin' -color red -command 'q_add_n -50 -2.5 0 .5 .05 0 0 0 2000 | q_sin' -color blue -command 'q_add_n -50 0 1.9 -2.3 .05 0 0 0 2000 | q_sin' -color green -command 'q_add_n -50 2 .8 2.2 .05 0 0 0 2000 | q_sin' -color orange -command 'q_add_n -50 1.5 -1 -1.8 .05 0 0 0 2000 | q_sin' -color black -command 'q_add_n -50 -1.5 -2.4 0 .05 0 0 0 2000 | q_sin' -color aqua -command 'q_add_n -50 -1.8 1.5 1.6 .05 0 0 0 2000 | q_sin' -color purple
math
equation:

$\sin(t,\vec{R}) = (\sin(t) \cosh(|R|), \cos(t) \sinh(|R|) \frac{\vec{R}}{|R|})$

tags
Physics Tag:
simple harmonic oscillator
Math Tag:
trig functions
sine
Programming Tag:
command line quaternions