Some interesting curves in the 3D space.

sine curve in a circle, sage code

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parametric_plot3d((cos(t), sin(t), 0.2*sin(15*t)), (t, 0, 6.28), aspect_ratio=(1, 1, 1), plot_points=200)

This can be parametrized like this:

\begin{align*} \begin{pmatrix} r \\ \theta \\ z \\ \end{pmatrix} &= \begin{pmatrix} 1\\t\\ \sin kt \\ \end{pmatrix} \end{align*}Translated into xyz-coordinates:

\begin{align*} \begin{pmatrix} \cos t \\ \sin t \\ \sin kt \\ \end{pmatrix} \end{align*}where t is the angle turned on the xy-plane.

spring in a ring, sage code

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parametric_plot3d(((5+cos(15*t))*cos(t), (5+cos(15*t))*sin(t), sin(15*t)), (t, 0, 6.28), aspect_ratio=(1, 1, 1), plot_points=200)

This curve is more complex, but can be simplified by using a different coordinate system.

- $r_1$: radius of the torus
- $\theta_1$: angle on the xy-plane
- $r_2$: radius of the tube of the torus
- $\theta_2$: angle on the intersection of the torus

Here is the lateral view. $\theta_1$ is invisible here.

Then we can parametrize it this way:

\begin{align*} \begin{pmatrix} r_1 \\ \theta_1 \\ r_2 \\ \theta_2 \\ \end{pmatrix} &= \begin{pmatrix} r_1 \\ t \\ r_2 \\ kt \\ \end{pmatrix} \end{align*}Translated into xyz-coordinates:

\begin{align*} \begin{pmatrix} (r1 + \cos kt)\cos t \\ (r1 + \cos kt)\sin t \\ \sin kt \\ \end{pmatrix} \end{align*}where t is the angle turned on the xy-plane.

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