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

iPod downloads:

sin-cos_xyz_constant.povray.100.1000.animation.scan_.m4v

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_xyz_constant.povray.100.1000.numbers