>A more interesting property is that any non-vertical line will intersect the curve in at most three places.

If this is correct, that means a line could intersect at 1,2, or 3 places.

>Let's imagine this curve as the setting for a bizarre game of billiards. Take any two points on the curve and draw a line through them; the line will intersect the curve at exactly one more place.

No that is not correct. I can already see two points where a horizontal line would only intersect 2 points and not intersect another. Basically, you can draw a line tangent to the curve on the left and intersect only 2 points instead of three.

These geometric explanations strike me as simplifications, and as I imagine what it would look like in code, I get the feeling that there would be all sorts of special cases, boundary conditions, and asymptotes.