If diverging rays never meet, why do parallel rays meet at infinity?

Question

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.

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

Answer 1

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.

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

"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).

Answer 6

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.