1. Introduction

Today’s post will be a little different from usual: I wanted to share a few of my exploits from long ago with the online graphing calculator Desmos. Specifically, I tried to use its features to design a few interactive tools that you can try out. I will supplement the link to each tool with a brief description of what it is and how it works. Enjoy!

2. The Cycloid Drawer

https://www.desmos.com/calculator/81brskzoje

The cycloid is a famous curve that can be generated by a simple process: take a circle with a marked point on its edge, roll the circle across a flat surface, and trace the path of your marked point. Although it seems like nothing beyond a recreational figure, the cycloid is a very mathematically and physically profound shape. Notably, it is the solution of the brachistochrone problem: given two points of different elevation, a ramp connecting them shaped like a cycloid will provide the minimum possible transit time between the points for an object sliding down the ramp. This is one of the simpler applications of Desmos: the cycloid itself has a nice parametrization obtainable by considering the rolling motion as a sum of translation and fixed-axis rotation. Other interesting curves can be generating by marking the point on the interior of the circle (or even outside it!); this is controlled by the parameter in the tool. Stay tuned for more rolling shapes and possibly an interactive demonstration of the sliding ramp problem…

3. The Involute Generator

https://www.desmos.com/calculator/y1xtzws3ms

I decided to make this tool after learning about an interesting concept in geometry: the involute of a curve. The involute of a curve is generated through a similar theme as a cycloid: take a line tangent to the curve and mark its initial point of tangency, then roll the line across the curve and trace the path of the marked point. Alternatively, this can be thought of as taking a string with a pencil at then end and wrapping it around the curve, then unrolling it while keeping tension. You may notice that any given curve actually has many different involutes, since one can begin the “rolling” process at different points on the curve:

4. Evolute Generator

https://www.desmos.com/calculator/p5ehkg5308

Finally, I will present the exact opposite of the involute generator: the evolute generator. The evolute of a given curve is the set of all centers of curvature of the curve. Specifically, for a fixed point on any curve (with nonzero second derivative), there is a circle in the plane that best approximates the curve at that point. The center of this circle is the center of curvature at that point, and the evolute plots all such centers of curvature. In this way, the evolute is actually the inverse operation of the involute: observe that when we generated the involute by rolling a line across the curve, we inadvertantly guaranteed that each point of tangency on the original curve would be the center of curvature of its corresponding point on the involute. In a later update, I may work on a version of the involute/evolute generators that accept general parametric curves rather than simple functions.