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I've seen that in the case of concave mirrors if the object is between focus and the pole - the reflected rays diverge and never meet.

enter image description here

But if the object is at the focus, it's defined to be meeting at infinity. Why is it so?

enter image description here

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    $\begingroup$ Believe it or not, this is the cause of huge stress to mathematicians and philosophers for thousands of years. It is closely related to (in fact is a different way of express) Euclid’s fifth postulate. en.wikipedia.org/wiki/Parallel_postulate $\endgroup$
    – J. Manuel
    Jan 3 at 8:09
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    $\begingroup$ @J.Manuel Actually in plane geometry we (mathematicians) do not say that parallel lines meet at infinity. For instance, nowhere in your linked article does it say that. That's something that does happen in projective geometry, but not strict Euclidian geometry. $\endgroup$ Jan 3 at 16:02
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    $\begingroup$ Perhaps a duplicate of this? physics.stackexchange.com/q/630986/221326 $\endgroup$
    – Draconis
    Jan 3 at 20:06
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    $\begingroup$ Someone should just delete the concept of infinity. $\endgroup$
    – Frank
    Jan 4 at 8:52
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    $\begingroup$ @RBarryYoung. I get you. That was an over-simplification and I was aware of that when making my previous comment. The sole Idea of that comment was to introduce the OP with the non-triviality of the problem he was facing, and maybe introduce him into the incredible fifth postulate problem which is undeniably connected to his own question, or “if they meet they are not parallel” (-Euckid's Fifth postulate). $\endgroup$
    – J. Manuel
    Jan 4 at 9:36

6 Answers 6

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If you align your viewing direction parallel to some set of parallel lines, you will visually see them ending at some "point" at infinite distance. The typical example is railroad tracks. railroad tracks meeting at infinity

If you take lines that are not parallel, then no matter what perspective you take, the visual point of intersection (if there is one) will always be a finite distance away, and thus not at infinity. E.g. the pole in the image is skew to the rails of the track, so no matter how you orient your view they will never appear to intersect at all, whether at infinity or not. Or take the rails and the wooden rail ties. They make right angles at points in real space, and no matter how you orient yourself you can never make them appear to intersect anywhere but at those points. Non-parallel lines are defined not to meet at infinity because our vision tells us they don't meet at infinity.

Also note that there are different points at infinity. The point at infinity at which the railroad tracks intersect is visually different from the one at which all the vertical lines in this photo intersect. And both are different from the one at which the horizontal wooden ties intersect. This is in disagreement with @nu's answer. This is because there are many ways to mathematically construct points at infinity given a suitable definition of "real space". My definition corresponds to projective space, instead of a one-point compactification.

The usage of many different points at infinity is justified by our visual intuition, and also by optical intuition. E.g. we usually idealize stars as point sources at infinity. But there are many stars that visually appear at different places in the sky. This is hard to make sense of if there is only one point at infinity, but if you instead construct many points at infinity, each star can get its own. Similarly, if you have a beam of parallel light rays and stick your eye in the beam, you will see the light as a "star" at the one point at infinity at which the parallel rays intersect, and not at a different point at infinity. If the rays instead intersect at some finite point, you will see a light source at that point, and not at any point at infinity.

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    $\begingroup$ This is an excellent answer $\endgroup$
    – Kai
    Jan 3 at 18:03
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    $\begingroup$ You get projective space if you add one point for each class of parallel lines, but it's also reasonable to instead add a point at infinity for each class of parallel rays (so two points at infinity on each line, one in each direction) in which case you get a sphere at infinity and a disk altogether. $\endgroup$
    – ronno
    Jan 4 at 10:24
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    $\begingroup$ Mmm.. Sorry, I don't like it. If your head was so wide that your eyes were the same distance apart as the rail tracks were and you focused your eyes to infinity so that each eye was looking straight down each rail, the rails wouldn't look like they come together. By extension of the same, let's have rails that gradually get further apart, so in ten miles down the track they're one inch further apart. The photo will look virtually the same (with our normal sized head and eyes aligned inwards), but we know the lines are non parallel yet our vision (like this photo) would say they do meet.. $\endgroup$
    – Caius Jard
    Jan 4 at 13:21
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    $\begingroup$ @CaiusJard Here is a quick demonstration of the fact that when you look straight down one rail, you do see the other rail coming to meet it, and of the fact that an ideal eye can see that slightly non-parallel lines do not intersect at infinity. $\endgroup$
    – HTNW
    Jan 4 at 17:00
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    $\begingroup$ @CaiusJard This converging effect has nothing to do with binocular vision. The picture in this post was taken by a monocular camera. $\endgroup$ Jan 4 at 19:45
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Infinity is not a real distance or an actual number. It's used in mathematics when describing limits as a parameter increases without bound.

Parallel lines, by definition, never actually meet in a flat plane (there are non-Euclidean geometries where they do meet, and these are relevant when General Relativity is taken into effect, but not for classical physics of light rays -- we can approximate space as a flat plane).

The distance from the mirror to the point where the rays meet is a function of the angle between the rays. The smaller the angle, the further the distance. Since angles can get infinitessimally small (ignoring Quantum Mechanics), this means that the distances can get infinitely large. Parallel lines have an angle of 0, so the limit of the distance as the angle approaches 0 is infinity.

In the mathematics, you'll have an equation with the angle in the denominator of a fraction. Dividing by 0 has no actual meaning in arithmetic, so that's why we use limits to deal with it.

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  • $\begingroup$ "there are non-Euclidean geometries where they do meet, and these are relevant when General Relativity is taken into effect, but not for classical physics" Parallel lines can meet on the surface of a sphere or spheroid such as the one we are living on. $\endgroup$
    – JimmyJames
    Jan 5 at 15:19
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    $\begingroup$ @JimmyJames Right, spherical geometry is non-Euclidean. $\endgroup$
    – Barmar
    Jan 5 at 15:19
  • $\begingroup$ I think then it's a little much to say it's not relevant to classical physics. We can only approximate the surface of earth as a plane up to a certain scale. $\endgroup$
    – JimmyJames
    Jan 5 at 15:24
  • $\begingroup$ I clarified that I meant physics of light in space, which approximates a plane. $\endgroup$
    – Barmar
    Jan 5 at 15:25
  • $\begingroup$ Better. Slight nit: I think maybe the right term here is 'hyperplane'. OK, my pedantic is showing. $\endgroup$
    – JimmyJames
    Jan 5 at 15:32
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In Euclidean geometry, parallel lines never meet. This is the very definition of parallel. So if the object is at the focus, the reflected rays indeed will never meet (in an ideal Euclidean world).

So why do we say they "meet at infinity"?

It turns out, it's just a notational convention. To borrow from another answer of mine:

When physicists say something "goes to infinity", what they mean is "as you take the limit, this value gets bigger and bigger without any bound, and will eventually exceed any number you choose".

In the standard system of real numbers (which is used for most things in classical physics), infinity isn't actually a number; it's more like a notational shorthand. So a more technically accurate way to say this would be:

As the object gets closer to the focus, the image (where the rays meet) gets farther and farther away, without any bound. You can make the image be as far away as you want, by bringing the object close enough. When the object is exactly at the focus, the rays are parallel, and thus never meet.

"The rays meet at infinity" is just shorthand for this.

Now, sometimes these sorts of things are modelled in projective geometry, rather than Euclidean geometry. And in projective geometry, "infinity" is actually a well-defined thing, and parallel lines actually do intersect at infinity. But from the wording of your question, I'm guessing you haven't been introduced to projective geometry yet; introductory classes tend to stick to nice, familiar Euclidean geometry, where "infinity" is just a nice bit of syntactic sugar.

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The fact that parallel lines meet at infinity becomes quite intuitive when thinking about what "infinity" actually means in a 2d plane. While the real numbers $\mathbb R$ are often compactified using two points, namely $+\infty$ and $-\infty$, to preserve their ordering in the compactification, in 2 dimensions, ordering does not make much sense (is $(2,1) > (1,2)$?), and a different compactification (the Alexandroff one-point compactification) is common, which only adds a single point, $\infty$.

This compactification can be pictured as follows:

  1. Identify the plane to compactify with the $x$-$y$-plane and add a third coordinate $z$.
  2. Place the center of a unit sphere at $(0,0,1)$, so that it touches the origin of the $x$-$y$-plane.
  3. Connect every point in the plane with $(0,0,2)$, which is the topmost point of the sphere, using lines, and identify the point where a line intersects with the plane with the point where the same line intersects with the sphere. This mapping $p: \mathbb R^2 \rightarrow \{\vec r \in \mathbb R^3 : |\vec r - (0,0,1)| = 1\} \setminus \{(0,0,2)\}$ is continuous and bijective and known as the stereographic projection.
  4. Add the point $(0,0,2)$ to the codomain and the point $\infty$ to the domain of the mapping and define $p(\infty) = (0,0,2)$. This definition makes sense, because for every sequence $(a_n)_n$ in $\mathbb R^2$ with $a_n \rightarrow \infty$ as $n \rightarrow \infty$, it obviously holds $p(a_n) \rightarrow (0,0,2)$.

Using this definition of infinity, it is clear that any two parallel lines both contain the single point $\infty$ and thus meet there.

Edit: Because the OP suggested the answer is too complicated, here are some additional explanations:

  • In this context, "compactification" can be thought of simply as "adding points at infinity". Whether or not the set is compact is not important for getting the general idea.
  • The codomain $\{\vec r \in \mathbb R^3 : |\vec r - (0,0,1)| = 1\} \setminus \{(0,0,2)\}$ is the sphere from 2. without the topmost point.
  • That the mapping in 3. is continuous and bijective means that it preserves the parts of the structure of the mapped plane we care about, namely that points which are "next to each other" stay that way. The problem with simply saying "points next to each other" is that this is not so easy to define for real numbers, as between any two of them there are infinitely many more.
  • Like Koschi explained in the comments, all infinite lines meet at the point $\infty$. They get mapped to circles containing $(0,0,2)$ on the sphere. The circles corresponding to parallel lines only touch at that point. However, if two circles intersect there, they have to do so at another point on the sphere, which will be mapped to a finite point on the plane.
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    $\begingroup$ If you don't mind, can you dumb down this a bit? $\endgroup$
    – Fr0zen
    Jan 3 at 8:28
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    $\begingroup$ I will try: Look at the stereographic projection of the globe/Earth with the north pole in the middle and Antarctica on the 'rim' (the UN logo looks like this, but is missing Antarctica). In this projection it is impossible to map the whole Earth including the south pole to a finite plane on this map, Antarctica becomes a large ring on the edge and the actual south pole would be infinitely far away, to all directions in the $\mathbb{R}^2$ plane... $\endgroup$
    – Koschi
    Jan 3 at 12:17
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    $\begingroup$ If you draw two parallel lines on this map, i.e. the 2D plane (parallel on the plane, NOT on the globe), they will never meet on the plane, but if you look how these parallel lines look 'projected back' to the globe it looks as if they meet on the south pole, which on the plane is projected to infinity. Note that also non parallel lines meet at infinity, i.e. look like they meet at the south pole, but they of course also meet somewhere else at a finite point in the plane, i.e. on a point that is not the south pole if projected back to the globe. $\endgroup$
    – Koschi
    Jan 3 at 12:17
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    $\begingroup$ Real projective space is probably a better answer -- the single point compactification makes all lines meet at the single point, where RP adds a point for each class of parallel lines. $\endgroup$
    – Spencer
    Jan 3 at 15:59
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    $\begingroup$ I feel like this is the wrong way to treat points at infinity in optics. You are describing a way to add a single point at infinity at the "end of all space". But optically the intuitive way to do it is to add many points at infinity, one for every direction possible in space, so that parallel lines meet at one point at infinity, and a different set of parallels will end at a different point at infinity. $\endgroup$
    – HTNW
    Jan 3 at 16:11
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"Infinity" here is actually a shortcut for saying "grows larger than any value you can name when the conditions approach condition X"; that is, it describes the behavior of an iterative procedure or an algorithm rather than being a static number (sorry, I'm a programmer).

In this case the procedure is to make the angle between two lines that run through two points in the 2D plane smaller and smaller. When the points are 1m apart and the angle is 90°, the lines cross at a distance of 1/2m. When the angle gets smaller, the crossing point moves farther away; there is no distance one can name that couldn't be exceeded by making the angle just a wee bit smaller. This is what we mean when when we say "parallel lines meet in infinity": The distance of the crossing exceeds any limit when the angle approaches 0 (i.e., when the lines become more and more parallel).

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The natural home for the geometry of plane curves is the projective plane, where everything is really much simpler. For example, a curve of degree $n$ and a curve of degree $m$ always meet in exactly $mn$ points in the projective plane (with a few provisos about exactly how to count), which turns out to be extremely convenient.

Lines are curves of degree 1, so two lines meet at exactly one point. The lines are called parallel if the line at infinity passes through that intersection point. But the "line at infinity" depends on your coordinate system, so it makes no sense to ask whether two lines are parallel until you've chosen coordinates. The same pair of lines can be parallel in one coordinate system and not in another.

When you work in the affine (euclidean) plane, you are choosing a line at infinity and throwing it away. Therefore lines that met at infinity (i.e. parallel lines) no longer meet at all.

Likewise (and not directly relevant to your question, but as another illustration of how the affine plane throws away information), a conic (that is, a curve of degree 2) is called a circle if it passes through two particular "circular points" at infinity. Two circles are called concentric if they are tangent at both of those circular points (here a tangency counts as two meetings, so the two tangencies use up all four of the intersection points). But again, the identity of the circular points depends on your coordinate system, so that whether a conic is a circle, and whether two circles are concentric, depends on your coordinate system. And if you throw away the line at infinity, concentric circles don't meet at all.

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