Many interesting curves can be represented in parametric form - meaning that the position of any point (in 3D space) on the curve can be expressed in terms of a single parameter (often denoted \(t\)). By varying the value of \(t\) within the animation and smoothly connecting the generated points we can trace out a curve in 3D space.
I created this visualization during my time lecturing multivariable and vector calculus at UCSD. I noticed that students often had a difficult time connecting abstract parametric equations to the geometry of the resulting curves. Allowing students to actually see the results of the parametic equations increased student engagement with the mathematics they were studying.
There are a variety of different curves to visualize:
\begin{align} x &= \cos(t)\\ y &= \sin(t)\\ z &= t \end{align}
\begin{align} x &= (R+a\cos(\omega t))\cos(t)\\ y &= (R+a\cos(\omega t))\sin(t)\\ z &= a\sin(\omega t)\\ \end{align}
\begin{align} x &= \cos(t)(R+a\cos(\omega t))\\ y &= \sin(t)(R+a\cos(\omega t))\\ z &= ht + a\sin(\omega t)\\ \end{align}
\begin{align} x &= t\cos(t)\\ y &= t\sin(t)\\ z &= t\\ \end{align}
\begin{align} x &= \frac{\cos(t)}{\sqrt{1 + a^2t^2}}\\ y &= \frac{\sin(t)}{\sqrt{1 + a^2t^2}}\\ z &= \frac{-at}{\sqrt{1 + a^2t^2}}\\ \end{align}
\begin{align} x &= \cos(t)\cos(t)\\ y &= \cos(t)\sin(t)\\ z &= \sin(t)\\ \end{align}
\begin{align} x &= \sin(t)+2\sin(2t)\\ y &= \cos(t)-2\cos(2t)\\ z &= \sin(3t)\\ \end{align}
\begin{align} x &= \alpha\sin(\beta t)\cos(t)\\ y &= \alpha\sin(\beta t)\sin(t)\\ z &= \sin(\beta t)\\ \end{align}
\begin{align} x &= t\\ y &= t^2\\ z &= t^3\\ \end{align}
\begin{align} x &= \cos (t) \left(a - \cos(\theta)\left(b-c\sin\left(\frac{at}{b}\right)\right)\right) - c\sin(t)\cos\left(\frac{at}{b}\right)\\ y &= \sin (t) \left(a - \cos(\theta)\left(b-c\sin\left(\frac{at}{b}\right)\right)\right) - c\cos(t)\cos\left(\frac{at}{b}\right)\\ z &= \sin (\theta) \left(b - c \sin\left(\frac{at}{b}\right) \right)\\ \end{align}