Some Insight on Differential Geometry

Suppose that you have a circle rolling on a flat surface without slipping and that it also has a dot on its edge. What path does the dot trace?

Tackling this question by means of the conventional, functional approach $y=f(x)$ is a tedious task. For that reason, we parameterize the function in terms of a convenient variable $t$ as such: $r(t)=g_1(t)\partial_x+g_2(t)\partial_y$. This is akin to a vector function.

Visually, this is the problem:


Taking the angle $t$ that the circle makes with its bottom-half vertical from its radius yields the following parameterization:$$r(t)=R\left[t+\cos\left(t-\frac{3\pi}{2}\right)\right]\partial_x-R\sin\left(t-\frac{3\pi}{2}\right) \partial_y.$$Plotting this parametric equation yields the following graph:

This is the path that the dot traces. Now, a good question could be, what exactly happens at the cusp? Also, what would happen if we slide the dot upward or downward from the circumference?

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