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:$$r(t)=R[t+\cos (t-\frac{3\pi }{2})]{\partial }_x-R\sin (t-\frac{3\pi }{2}){\partial }_y.$$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?