In classical geometry why is a line considered to be parallel to itself?

Question

A definition in classical geometry (for example, Birkhoff's formulation, but I suppose it could be all of them) is that a line is always considered to be parallel to itself. I understand this is probably for convenience, but in my mind since two distinct lines are parallel if they have no points in common and a line has infinitely many points in common with itself. Perhaps the idea is to ease the definition that two (non-parallel) lines intersect at one and only one point?

Q: What's the purpose/what inconvenience would be caused if we didn't have that definition?

Answer 1

The idea is that you want "parallel" to define equivalence classes (called "pencils", cf. Coxeter, Projective Geometry, and Artin, Geometric Algebra), which require the defining relationship to be an equivalence relationship: reflexive, symmetric, and transitive. Those classes then have some nifty uses, like defining projective space by adding a point at infinity for each pencil (which is the considered to be on each of those lines) and a line at infinity for each class of parallel planes (this line containing all the points at infinity corresponding to the pencils of lines in that class of planes).

Also, you were already going to have to rethink the definition of "parallel" as having no points in common, if you are going to do solid geometry. Parallel lines also need to be coplanar...i.e., there need to be two other lines that intersect each other and that each intersect the parallel lines (five distinct points of intersection). Lines that are not coplanar are called "skew" not "parallel".

Answer 2

If two lines are both parallel to a third line, they should be parallel to each other, even if they are the same line.

Answer 3

In .. "parallel" to the answers already given, defining that a line is parallel to itself has the advantage to match, on the algebraic side, the distinction of linear systems based onto the coefficient matrix and the total matrix.

So we define parallel the lines for which the coefficient matrix has rank $1$ ( thus they have parallel normal vectors), and then we distinguish them as "distant" (no solution) or "same"(infinite sol.) according to the rank of the total.

Answer 4

It is a (useful ?) convention.

According to Euclid's original definition two parallel lines must be different:

Definition 23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Thus parallelism is not reflexive.

The same with Hilbert's version :

Axiom III. In a plane $\alpha$ there can be drawn through any point $A$, lying outside of a straight line $a$, one and only one straight line which does not intersect the line $a$. This straight line is called the parallel to $a$ through the given point $A$.

We may compare with Birkhoff's definition :

In consequence of Postulate II, any two distinct lines $l, m$ have either one point in common or none. In the first case they are said to intersect in their common point; in the second case, they are said to be parallel; a line $l$ is always regarded as parallel to itself.

Answer 5

This is also quite natural when starting from the definition in a metric space:

Two lines $a$ and $b$ are parallel if for every point $p\in a$, the distance from $p$ to $b$ is the same.

Then, $a$ is parallel to itself because for every point $p\in a$, the distance to $a$ is zero.

Answer 6

When comparing different functions, often points of interest are where the slope of the graphs is the same: parallel.
It would be quite inconvenient if two slopes would be considered "not parallel" just because they're on the same line.

Answer 7

The definitions of parallel vectors:

Two vectors $u$ and $v$ are parallel if their cross product is zero, i.e., $u\times v=0$. Two vectors are parallel if they are scalar multiples of one another.

Similarly parallel lines can be characterized as:

Two lines are parallel if they are scalar additions of one another.

Obviously, in this sense the equal vectors as well as the same lines are parallel.