In high-school I learned how to find the circles that connect points in 2D space forming a curve made out of tangent circles like this:
(The green line shows the "initial direction" of the curve)
You could imagine trying to close the curve with one last circle. However, this doesn't always close the curve "nicely":
And sometimes it does:
My question is what conditions must these points follow such that they can be connected with a "nice" curve?
What I have found messing around is that, for an odd number of points, it seems to be always possible to find an "initial direction" that closes the curve nicely. But for an even number of points, this is only possible for some set of points: Sets that do close nicely for one "initial direction", close nicely too for all "initial directions". For example, points that can be all put in one circumference always close nicely.
I think I have some intuition for what is going on but I can't find a nice mathematical answer. I also can't find anything like this on the internet, so I'm sorry if this isn't very rigorous. Is there an official name for this kind of curves? Where could I find more information and theory on this?
Best Answer
The comment by mlk is right on the mark. The idea of this solution is basically what he wrote, but there a lot of messy details coming from the question of which direction the arcs are curving. Let the points be $P_1$, $P_2$, ... $P_n$ in that order, and assume that the curve is a simple closed curve. Let $\alpha_i$ be the angle $\angle(P_{i-1} P_i P_{i+1})$, measured inside the simple closed polygon. I will show that:
For $n=2m$ even, the curve closes if and only if $\alpha_1+\alpha_3+\cdots+\alpha_{2m-1} = \alpha_2 + \alpha_4 + \cdots + \alpha_{2m}$. In this case, we can start at any angle. (Since the sum of the angles of a $2m$-gon is $(2m-2) \pi$, the common value of the sum of $(m-1) \pi$.)
If $n$ is odd, there is exactly one angle we should start at.
Let $O_i$ be the center of the arc from $P_i$ to $P_{i+1}$. I found the following sign convention useful: If $O_i$ is inside the curve, I set $\theta_i = \angle(O_i P_i P_{i+1}) = \angle(O_i P_{i+1} P_i)$; these angles are equal because they are the base angles of an isosceles triangle. If $O_i$ is outside the curve, I put $\theta_i = \pi-\angle(O_i P_i P_{i+1}) = \pi-\angle(O_i P_{i+1} P_i)$.
The condition that the arcs $P_{i-1} P_i$ and $P_i P_{i+1}$ are tangent says that $O_{i-1}$, $O_i$ and $P_i$ are collinear with $O_{i-1}$ and $O_i$ on the same (respectively opposite) side of $P_i$ if $O_i$ and $O_{i+1}$ are on the same (respectively opposite) side of the curve.
I claim that the following relation holds: $$\alpha_i = \theta_{i-1} + \theta_i. \qquad (\ast)$$ We need to check all four cases for whether $P_{i-1}$ and $P_i$ are inside or outside of the curve: For example, if both are inside, then this holds because $$\angle(P_{i-1} P_i P_{i+1}) = \angle(P_{i-1} O_{i-1} P_i)+ \angle(P_{i} O_{i} P_{i+1}) = \angle(P_{i-1} O_{i-1} P_i)+ \angle(P_{i} O_{i+1} P_{i+1}).$$ In the other cases, we need to similarly check that all the $\pi$'s and $-$'s work out.
We can use $(\ast)$ to compute all of the $\theta_i$ in terms of $\theta_1$: We have $\theta_2 = \alpha_2-\theta_1$, $\theta_3 = \alpha_3-\alpha_2+\theta_1$, etcetera.
If $n$ is even, we see that the curve closes if and only if $\alpha_1+\alpha_3+\cdots+\alpha_{2m-1} = \alpha_2 + \alpha_4 + \cdots + \alpha_{2m}$. If $n$ is odd, we see that the key equation is $\theta_1 = \alpha_1 - \alpha_2+\alpha_3-\alpha_4 +\cdots +\alpha_n - \theta_1$. This will hold exactly for one value of $\theta_1$ modulo $\pi$. $\square$
I tried to work out the case of a self intersecting curve, but it got too messy. There is a method called "directed angles" which often cleans up situations like this -- see Section 1.7 of Geometry Unbound -- but I didn't find it helpful.