The following puzzle occurred to me:
Suppose we want to devise a code such that given any line in the plane, you can give me $k$ real numbers, and from this information I can uniquely determine the line. What's the smallest $k$ can be?
For example, you could specify 4 numbers listing the coordinates of two points on the line. In fact, we can have $k=1$ since we know there exists an injection from $\mathbb{R^4}\to\mathbb{R}$. However, the nicest geometrically inspired map I could think of has $k=2$. So, I have two questions:
Is there a simple (i.e., geometric and intuitive to a high schooler) way to uniquely determine a line with a single number?
How about the analogue for circles? We can specify 3 numbers (center coordinates and radius), but can we get it down to 2 or 1?
For (1), the moduli space of all (unoriented) lines is given by the Möbius strip, so you need at least 2 parameters unless you want to break continuity. (The description is of course $(\theta,d)\mapsto(x\cos\theta+y\sin\theta=d)$ with $(\theta+n\pi,(-1)^nd)$ for all $n\in\mathbb{Z}$ describing the same line.)
(2) is similar, you have already shown the moduli of circles is 3-dimensional.