1

A 2D euclidian plane R2 contains a line and a ray.

The line is vertical and positioned where $x = 0$. The point on the line where $y = 0$ is labelled $m$.

One end of the ray is at $(x, y) = (1, 0)$ pointing in the negative $x$ direction.

The ray intersects the line at a distance along the line of $n$ from $m$. Initially $n = 0$ but when turning the ray the value of $n$ changes.

When rotated far enough the ray becomes parallel to the line. As we tend towards this configuration $n$ tends towards infinity.

I've heard the stronger term used that "parallel lines meet at infinity" even outside of the context of projective geometry. As infinity is a single value at both ends of the real number-line, presumably with parallel lines all 4 line ends in question meet at that infinite point.

Q: Returning to the case with a line and a ray, is it ever well defined and useful to consider (name and use) the value of $n$ reached when the ray is pointing away from the line? For example along the $x$ axis in the positive direction.

alan2here
  • 1,037
  • 1
    Did you mean the ray starts at $(x,y)=(1,0)$? – Daniel Mathias Dec 31 '18 at 17:50
  • 1
    This might not be what we're looking for, but certainly the asymptotic behavior of functions tending to infinity - especially the comparative asymptotic behavior - can be quite interesting. Specifically, what you have in this case is (for example, framing things in terms of rotation) a function from $[0,{\pi\over 2})$ to $\mathbb{R}$ sending a given angle to the distance from the origin of the resulting "meeting point." Rotating at a different speed, or using a different ray, or etc. will change the behavior of the resulting function. (cont'd) – Noah Schweber Dec 31 '18 at 17:57
  • 2
    The key point here is that we don't want to distinguish between the "infinities" these functions tend to (indeed, under the most natural interpretation they're all the same "infinity"!), but rather the manner in which they tend to infinity. (If you really want to "assign an infinity" to a function, you can do something like this via nonstandard analysis, but all you're really doing is rephrasing the asymptotic behavior.) I don't know about the specific example you're describing, but certainly general asymptotic analysis is quite important. – Noah Schweber Dec 31 '18 at 18:00
  • Thanks Daniel. I've now fixed the typo :-/ I'm just reading Noah's comment :) – alan2here Dec 31 '18 at 18:14
  • https://math.stackexchange.com/questions/661855/is-it-faster-to-count-to-the-infinite-going-one-by-one-or-two-by-two/662220#662220 – Asaf Karagila Dec 31 '18 at 18:19
  • You might find it useful to read up on one-point compactification. – amd Jan 01 '19 at 00:35

0 Answers0