Parametric paths

Here are a couple of dynamic visualizations that illustrate how circles and lines can be mixed and matched to powerful effect.

A cycloid

A point on a rolling circle traces out a path called a cycloid. The simplest form of a cycloid is \[ \begin{aligned} \vec{p}(t) &= \langle t-\sin\left(t\right), 1-\cos\left(t\right) \rangle \\ &= \langle t,1 \rangle - \langle \sin(t), \cos(t) \rangle. \end{aligned} \] Note how the vector formulation shows how the path arises as the combination of a linear motion and a circular motion.

A helix

A helix also arises as the combination of a linear motion and a circular motion. This time, though, the linear motion is orthogonal to the plane of the circular motion.