I want to view what the graph of, for example,
$x=\sin t,\quad y = \sin(10t)$
looks like, but not as a static graph $\ y=f(x),\ $ but rather one where we can see the movement of the point on the $\ x-y\ $ graph
$(\ x(t),\ y(t)\ ) = (\ \sin t,\ \sin(10t)\ )\ $
as $\ t\ $ varies, starting at a specified time, e.g. $\ t=0\ $ and then seeing as a video how this point moves with $\ t\ $ increasing from $\ t=0.$
I'm sure I could program this with Python or something, but is there already an option for this with the free versions of Desmos or Mathematica/ WolframAlpha?
Best Answer
Desmos can handle this for you. Here's an example of the parametric equation that you've described: https://www.desmos.com/calculator/fcivlr8pll
Here's a short description of the code as well.
Writing a pair of parametric equations $(x(t), y(t))$ in Desmos will automatically parameterize the curve for $t\in[0,1]$. This will be a static curve, but you can make it dynamic by including another parameter (I call it $s$ in my program) and reparameterizing $(x(t), y(t))$ for $t \in [0,s]$. Finally, you can demonstrate where a particle is along this trajectory by including the line $(x(s), y(s))$.
I find Desmos incredibly handy for visualizing paramtric equations. If, for example, you want to study parametric equations in the complex plane, I would suggest using Desmos. Here's a more complicated example of what Desmos can handle: https://www.desmos.com/calculator/6o1a3cx47b.