# How can parallel rays meet at infinity?

I found that in every book (till my 12th) it is written that, in concave mirror, when object is at focus, then reflected rays will be parallel and they meet at infinity to form a real image.

But, as we know, parallel rays never meet. Then, does this mean that all books are wrong ? If not, then why?

• As well as the answers below, you may find this post useful physics.stackexchange.com/questions/155996/…. Apr 21, 2021 at 14:25
• They meet at infinity, but by definition they never get there so they never meet. Apr 21, 2021 at 19:15
• "Infinity" isn't a real number, or a place. Think of "meet at infinity" as a kind of shorthand -- although exactly what it's a shorthand for is subtle, and is best understood by reading about projective geometry -- but rays that "meet at infinity" will in fact never meet. Apr 21, 2021 at 22:16
• I would suggest to ask this question at math.stackexchange to get (more) coherent answers.
– user296951
Apr 21, 2021 at 22:21
• See the Wikipedia article on Projective Space; for intuition, it's just a useful concept when it comes to projecting rays; imagine shooting a ray at some point that's very far (e.g. the center of the Sun); move some distance away, aim at the same point & shoot another ray - the resulting lines will be essentially parallel. In the limiting case, when the target is at infinity, there will be one point "at infinity" for every direction, and all parallel lines going in that direction "meet" there. Apr 22, 2021 at 7:32

It means that they don't meet, because as you correctly pointed out parallel lines never meet.
Then what's the point in saying "they meet at infinity" if they never meet? Because you can obtain a parabola by an ellipse with focal distance $$d$$ in the limit where $$d\rightarrow\infty$$. In the ellipse rays from one focus get reflected to the other one, and the same happens for a parabola, but one focus is at infinity (therefore you'll never reach it, and therefore the rays won't meet).

Edit: since many are mentioning it in the comments, I mean in $$\mathbb{R}^n$$, where Euclidean geometry is safe. There surely are more interesting spaces, but I thought OP was referring to the flat, euclidean case (and also, in our everyday lives, it's the space where the physics of reflection lives).

• They of course meet. The point at which they meet, is an imaginary point, that is all. It follows from fairly simple trigonometry that the only solution for the intersection is the point $\infty$. Furthermore, it is a nice definition of parallel, since that property would only be held by a parallel line. Parallel lines meet at infinity. This definition holds as you add dimensions, since there are skew lines, that never meet, but aren't parallel. Apr 22, 2021 at 10:34
• @StianYttervik if you work in R2, as I assumed OP was, then the parallel lines don't meet. "parallel lines meet at infinity" in Rn is an abuse of language: one can say it and mean that they would meet at the compactified infinity on the Riemann Sphere (or any other equivalent description of a compactified infinity point) but there's no such point in Rn. Also, there are no imaginary points in this argument: there is a perfectly real point on the Riemann Sphere, and there isn't one in Rn. Apr 22, 2021 at 10:54
• You are right. I was a bit direct and maybe obstinate - but my point is that this all resolves nicely, and fits with trigonometric identities and equations if one just accepts the nature of the point inifinity. Once this is accepted, the intuition sits fairly well (that parallel lines meet there, and only there) and it also scales into higher dimensions. It nicely fits into parabolic definition, a parabola which is focused at infinity, collects the rays that meet at infinity. It is the intuition of the point of infinity which the OP needs to resolve. I think. Apr 22, 2021 at 11:04
• @MauroGiliberti Just a quick nitpick: the one-point compactification of the complex numbers, otherwise known as the Riemann sphere, actually doesn't guarantee that any two lines intersect. Rather, the space in which the statement OP is referring to is true is the projective space, which doesn't have one point at infinity, but many. Apr 22, 2021 at 14:49
• I think the connection between parallel lines and an "infinitely far" focal point is important. Yes, they never meet, but they never meet in a very special fashion that distinguishes parallel lines from other non-meeting lines (skewed, divergent). I think that was part of @StianYttervik' s gist. Apr 23, 2021 at 14:03

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.

EDIT: As Don Thousand points out in the comments, situations like this are sometimes handled in projective space, where infinity has a concretely defined meaning, with the projective real numbers (or projectively extended real numbers), where infinity is in fact a number. But in my experience, introductory texts tend to avoid this in favor of the Euclidean space and real numbers students are used to.

• I disagree. infinity in this sense is a bit more concrete than just syntactic sugar. In projective space (which is generally the space in which we make statements like this), there are actually points at infinity, at which parallel lines intersect. Apr 22, 2021 at 14:46
• @DonThousand True, but in my experience introductory books don't usually cover the projective reals. I'll add a note about that though. Apr 22, 2021 at 15:48

This statement is based on a mathematical construct known as projective space.

The idea behind projective space is this. Start with the usual Euclidean space, $$\mathbb{R}^3$$. Now consider an infinite straight line passing through the origin. (We don't need an origin, but it makes the construct easier to visualize.) There are, of course, no endpoints of this line. So what we do is we create some - we declare by fiat there now exist two points further along the line in either direction than any other points thereupon, i.e. further away than any real number distance. We call these two points the points at infinity, and we repeat this for every line through the origin. If you are following this correctly, you should imagine that, in a sense, the whole space gets surrounded by an infinitely large "sphere" of these newly-minted points.

The next step is to now identify the points at opposite ends of each such line, so that opposite points on the "sphere" become the same point, i.e. we now modify the space again so that if you're standing at the origin and you look straight ahead, you are looking at the same point as if you turned around and looked straight the opposite way.

The final step in the construction is perhaps the one that will seem the most dubious to you: consider a line not passing through the origin. What point(s) at infinity, if any, does it connect to? Well, we define this to be the same point at infinity that a line through the origin which is parallel to it connects to. When you do this, it necessarily becomes the intersection point of both those parallels, since it lies on both. It will follow then that any pair of parallels in any given direction will, by transitivity, intersect at the point at infinity that was defined as being in that same direction from the origin. (And thus my earlier statement that we can dispense with the origin.)

Why do we do this? The intuition here is this. If you look down a pair of extremely long parallel lines extending far from you, the two appear to your eyes like they would eventually converge at some point in the distance, even if they don't "in reality". The idea here is to take this apparent limiting point and make it real in the idealized mathematical world. Then the mathematical lines intersect there. Moreover, if you move laterally from side-to-side, so that you are looking down the parallels from a different "origin", the distant point never shifts.

When we do optics in physics, we then use lines in this projective space as a model for our light rays, and these modeled rays have the intersection you describe. It's not, of course, an exact model - physical light rays will never travel infinitely far - but on the other hand, no physical model is such or, at least, all physical models are either one of limited in validity or else not empirically verified to be truly exhaustive of reality. But in doing this, we can treat all intersection cases on an equal footing, thus simplifying the mathematical treatment of optics.

• Is it really based on projective space? Apr 22, 2021 at 16:58

The books are just suggesting that as the object distance approaches the focus (from outside of the focus), the image distance will approach infinity.

It might help to slightly rephrase it.

A Hey, these two lines are parallel!
B Indeed.
A I wonder where they meet?
B Follow the lines, and you will see them meet when you have travelled an infinite distance.
A But I will never complete a journey of an infinite distance!
B Exactly.

Or, maybe more tersely:

A How far must I keep travelling to see these two parallel lines meet?
B You will never see it happen, so your journey would be infinitely long.

In the example you've encountered, infinity is effectively equivalent to "N/A", i.e. meaning that you were expecting to see a fixed number here expressing a specific value, but no such value exists, so infinity is used to convey that message.

There are more mathematically inclined reasons for why we use the concept of infinity like this, but I infer that your question is more of a semantical nature whereby you read that "at infinity" implies a known location, and therefore cannot be nothing.

In a way, it's similar to saying "when pigs fly". If I tell you that "I'll give you a million dollars when pigs fly", that's just the same as me saying that "I'll never give you a million dollars".

Just because I say "when pigs fly" doesn't mean that I genuinely believe that pigs will one day fly. I'm actually relying on quite the opposite, that they never will fly.

Similarly, when I say that "two parallel lines meet at infinity", it's just another way of saying "two parallel lines never meet", because I am relying on the idea that you can never reach infinity.

Concave and convex mirrors are special cases, by definition different from plane mirrors.

Depending on the degree of curvature, the reflected rays might appear in the foreground with their true parallelity, but in the (far) background they will converge or diverge.

Even in a plane mirror, perspective will make parallel rays appear to meet in the background, just as they will when viewed directly.

There are many excellent answers already, but I think it may be worth giving an concrete example of how we can think of parallel lines intersecting at infinity via the Riemann Sphere.

First we need to know of stereographic project, that is mapping a sphere to a plane via stereographic projection. I found the best explanation of this in this answer. The basic idea involves, imaging that you are on top of a beach ball in the middle of the plane, and pointing at points on the plane through a laser which pierces through the surface of the sphere.

It may be seen that circles on the surface of the sphere gets mapped to lines on the plane as shown here:

And, intersections of lines corresponds to intersection of the circles on the sphere which pass through the north pole of the sphere. It is defined that the point at the north pole is mapped to infinity under the mapping, putting these two facts together, we can , in a way, think that the parallel lines do meet at infinity.

Picture from page-140 of Tristan Needham's Visual Complex Analysis and more technical details can be found on wikipedia

There are several meanings that can be ascribed to this. One is that the limit as we approach this situation of the intersection distance is zero. Another is that the formula for image distance outputs infinity.

Another is to treat optics as taking place in a projective space. The claim that parallel lines never intersect is true in standard Euclidean space, but in projective space, parallel lines do intersect (in the projective plane, lines are dual to points; just as every two points define a line, every two lines define a point). The point at which they intersect is often refer as being "infinity", but there are constructions of projective space that don't make any reference to "infinity". Also, different pairs of parallel lines intersect at different points, so if the points are referred to "infinity", it has to be understood that there are different "infinities".

Other answers have brought up projective spaces and mentioned that textbooks don't go into the rigorous mathematical foundation. This is generally the case in physics textbooks, especially those for earlier classes (I'm not sure what "my 12th" refers to; if it's 12th grade, projective geometry is generally considered too advanced for high school). Physics textbooks will interchange limits and integration without discussing their justification, speak of "infinitely long wires", etc.

Using mathematical formulations such as projective space where infinity is meaningful is very useful in optics because of how much it shows up. When we have parallel rays, we can diffract them through a lens, and we can use their original convergence distance of "infinity" in our lens formula to find where the new image will form.

tl;dr They're referring to a separation of $$\frac{1}{\infty}=  0 " ,$$ rather than an actual zero. Since there's an infinitely small defect in the rays being parallel, they can intersect at an infinite distance. By contrast, if the object were exactly at the focus, then the rays would be perfectly parallel and not intersect.

### The rays have an infinitely small deviation from being perfectly parallel, allowing them to intersect after infinite distance.

How can parallel rays meet at infinity?

They can meet at infinity in the same way that two different points can be separated by zero-distance, as is the case in your book:

I found that in every book (till my 12th) it is written that, in concave mirror, when object is at focus, then reflected rays will be parallel and they meet at infinity to form a real image.

First, to be clear: if the object is at the focus, i.e. they literally coexist at the exact same point in physical space, then the lines don't meet.

However, your book appears to be referring to the case where the object and the focus are at different points separated by zero distance – the $$ \frac{1}{\infty} = 0 " \text{-zero} ,$$ that's infinitely small, rather than the integer-zero that's perfectly zero.

Technically they're being loose with the language – it's somewhat cutesy to make trippy statements about stuff "at infinity" by exploiting the ambiguity in that terminology – and referring to a distance that's zero only in the sense of lacking a finite difference.

In short, they're saying that if the defect in the rays' parallelism is infinitely small, then the rays can meet at an infinitely large distance.

### Discussion: Understanding the math.

If you want a good overview of infinities, then might want to read about the hyper-real number system. It's a generalization of the real number system that's often taught in schools and can be used in physics whenever you want. Moving from reals to hyper-reals is sorta like moving from 32-bit integers to 64-bit integers when programming.

But to use simpler notation, I'd suggest considering the proposition that $$\frac{1}{\infty} = 0 \,.$$ The error in this equality is infinitely small, and since it's infinitely small, the only way to blow it up is to do something that multiplies it by an infinitely large value.. which is usually taboo in early education.

But that's what's happening here: you've got two rays that approach each other by an infinitely small amount, such that they'd never meet over a finite distance. But when we start talking about infinitely-large distances, then it's sorta like saying \begin{align} \infty \times \frac{1}{\infty} &= \infty \times 0 \\ & \Downarrow \\ \frac{\infty}{\infty} &= 0 \,, \end{align} where the silliness of the proposition becomes apparent.

And that's when parallel lines start intersecting: because their infinitely-small approaches can add up over infinite distance.

It is easy to prove that the frequently heard statement 'Parellel lines meet at infinity" is mathematically incorrect:

A necessary condition for lines to meet is obviously that their distance $$d$$ is zero. But if you have two parallel lines along the x-direction a distance $$d=1$$ apart, then

$$\lim_{x\to \infty}d(x) =1$$

so the lines do not meet.