Is a circle just a line (therefore 1 dimension) or is it a 2-dimensional object because it occupies some surface?
Thanks in advance!
[Math] How many dimensions does a circle have
circlesgeneral-topologymanifolds
Related Solutions
If you draw the radius of the circle to the point of contact, the tangent is perpendicular to the radius. You can use that tangent as a flat surface to bounce off of, so the two angles you have drawn go from the green arrows to the tangent line and are equal. The problem is quite ill-conditioned. A small error in locating the impact point will make an error in the angle you calculate, which will propagate to the next collision with a lever arm of the free path. Maybe the ill conditioning doesn't matter too much for your application.
Use the Law of sines
Let $R$ the circumradius of the triangle $ABC$ \begin{align*} \frac{|AC|}{\sin B}&=2R\\[4pt] \frac{\sqrt{2^2+4^2}}{\frac2{\sqrt{2^2+16^2}}}&=2R\\[4pt] \frac{\sqrt{20}\sqrt{260}}{2}&=2R \end{align*}
Then $R=5\sqrt{13}$.
Best Answer
A circle is a one-dimensional object, although one can embed it into a two-dimensional object. More precisely, it is a one-dimensional manifold. Manifolds admit an abstract description which is independent of a choice of embedding: for example, if you believe string theorists, there is a $10$- or $11$- or $26$-dimensional manifold that describes spacetime and a few extra dimensions, and we can study this manifold without embedding it into some larger $\mathbb{R}^n$.
Incidentally, you might guess that $n$-dimensional things ought to be embeddable into $\mathbb{R}^{n+1}$ ($n+1$-dimensional space). Actually, this is false: there are intrinsically $n$-dimensional things, such as the Klein bottle (which is $2$-dimensional) which can't be embedded into $\mathbb{R}^{n+1}$. The Klein bottle does admit an embedding into $\mathbb{R}^4$. More generally, the Whitney embedding theorem tells you that smooth $n$-dimensional manifolds can be embedded into $\mathbb{R}^{2n}$ for $n > 0$, and for topological manifolds see this MO question.
There are also other notions of dimension more general than that for manifolds: see the Wikipedia article.