$\cos{x}$ graphed on $e^x$

One of my friends wondered aloud whether it was possible to graph one function on another function as if one function were the x-axis. We decided to work on a simple case with both input functions as single-variable functions. In this simplified case [1], it helped to think of one function as a series of vectors on another function. An essential part of the solution was finding the length of the axis function. At the time, I was in precalculus and did not know much about calculus, so I had to approximate the curve’s length. Luckily, I came across calculus of variations. Calculus of variations is usually introduced with the example problem of minimizing a path between two points, which used a calculus-based function to get the path length. This concept was exactly what I needed to solve the problem. After solving this simplified version, I generalized it for if the axis function were parametric [2], using the knowledge I had gained from BC Calculus.

$0.05\sin{500x}$ graphed on parametric $(\cos{3x}, \sin{4x})$