Concepts in Curvature

Step 1: Define the Curve

To find the curvature of a curve, we first have to define the curve! The curve must be a set of parametric equations that are defined in terms of $t$, where both $x(t)$ and $y(t)$ are twice differentiable.

That might sound complicated, but the best place to start is with something defined with sine and cosine. Here are the equations for the cycloid that we'll use throughout these steps, where $a$ and $b$ are constants.

This set of parametric equations will be referred to as $s$ or $s(t)$ during this tutorial. For the time being, we'll use $a=0.5$ and $b=1.5$. Give yourself a second to get familiar with this graph, and then move on to the next step.