Do parallel lines merge at infinity?

Question

My school mathematics book says that parallel lines are always apart by a fixed distance and the lines never intersect.

The rudimentary explanation provided is that the tangent drawn at any point on the lines is always 90 degrees.

I can cross verify it by drawing tangents to the lines at small fixed points, but what I don't get is how the proof implies it to even hold at infinity. The infinity is undefined and I think it's possible for mathematical operations to glitch at this point, for example, a division overflow when you keep on increasing n towards infinity.

while (n++) terminates when a large N-bit register holdings the value n overflows and returns to 0, which means that parallel lines merge/intersect at/post infinity.

Answer 1

Parallel lines don't meet. Ever. That's the definition of "parallel" so it does not need any explanation.

One of the axioms (assumptions) of Euclidean geometry is that there are parallel lines. Since that is an assumption, it can't be proved.

There are non-Euclidean geometries in which there are no lines through a given point parallel to a given line, and geometries in which there are many.

Similarly, it's one of the axioms of the integers that they go on forever. The fact that some computer implementations of an integer data type can overflow is a property of the implementation, not the integers. And there are ways to manage integers as large as you wish in computer programs without having to deal with infinitely many integers.

Answer 2

"Parallel lines don't meet. Ever." - Sure that's right. But it is right only within the Euclidean space, the coordinate plane associated to $\mathbb{R}^2$. But then too infinity is not a concept of $\mathbb{R}$, in fact $\mathbb{R}$ is defined to be an open set.

OTOH you could well consider compactifications $\overline{\mathbb{R}}$ of either dimension of $\mathbb{R}$, just as an open interval $]a,b[$ can be embedded into a closed interval $[a,b]$. So, there are various such compactifications. Just as $[a,b]=\{a\}\ \cup\ ]a,b[\ \cup\ \{b\}$ one possibility would be to consider $\{-\infty\}\ \cup\ \mathbb{R}\ \cup\ \{+\infty\}$, a different possibility would be the one point compactification of the real line (kind of closing up into a circle) $\mathbb{R}\ \cup\ \{\infty\}$.

The same then holds for the plane. Here you fix first a single point, say the origin. Then you consider the fan of all lines running through this point. For any such line you either add both line ends or you just add a single additional point to that very line. Both these concepts will add an additional "line of infinity", described by the set of all these added points. In the first case however, even so being locally 1-dimensional, it is not a line within the euclidean space sense. It neither is open ended nor it is a 2-point compactified line. Rather it is a full circle (with infinite radius, thus zero curvature). In the second case that added "line" however has right the same form as the extensions of the normal lines. In fact, this second way of embedding of the euclidean plane is the projective plane, where virtually any chosen such line could be considered as the line of infinity wrt. the remainder subspace, which then is just an euclidean space again.

Still there is a further way of compactification of the 2-dimensional real euclidean space. It is the 1-point compactification, as usually is chosen in complex analysis. You will not add a different single point of infinity for every direction, rather you would identify all those directional points instead as a single added point. Thus you just have $\mathbb{R}^2\ \cup\ \{\infty\}$ (kind of closing the plane into a sphere).

You mentioned that (in euclidean context) 2 parallel lines never intersect and that this is being "proved" by the fact that those 2 lines always will maintain the same distance apart. But you will have to note here however that any such distance measure is subject to the corresponding definition of measure of the respective space. E.g. the distance between the (euclidean) points $(a,0)$ and $(0,a)$ clearly is $\sqrt{2}a$, which thus obviously runs towards infinity too, whenever $a$ runs to infinity. But within 3-dimensional embedding of that 1-point compactified plane, i.e. that sphere, any line in the plane is nothing but a circle on the sphere which runs through that additional point $\{\infty\}$. Thus the 3-dimensional embedding space measure clearly would imply the wrong distance measure for those mentioned points! Rather you have to stick to the appropriate measure definition of your base space instead (at most extended to support those added points somehow).

--- rk

Answer 3

"Do parallel lines merge at infinity?"

The answer of "No" was not intuitively evident to some ancient geometers who were familiar with lines that "meet" at infinity (i.e, "asymptotic lines.")

Euclid had defined parallel lines to be straight lines in a plane that "being produced indefinitely in both directions" never intersect; and accordingly, will never meet (or "merge") even at infinity.

So, if you are in the realm of Euclidean geometry, then parallel lines can never intersect, even at infinity.

However, there are non-Euclidean geometries where the Parallel Postulate has been modified in different ways to produce many counter-intuitive results. But I am presuming you are referring to Euclidean geometry where the Parallel Postulate is an axiom; i.e., a truism which has no proof.