# How do you call two lines who share an infinite number of points?

There's a word in French for when two lines share an infinite number of points. They're superimposed, and ultimately the same line, but if you see them as two lines, is there a word to describe them?

This poses no problem at all in French but according to this question, the concept doesn't seem to exist. The French word is "confondu" and should be understood as "indistinguishable", "merged".

That also mean that in the definition of parallel lines, we also consider the case where the lines are confondues. If they share every point,

Here's an example:

Let $$l$$ be a line and $$A$$ a point on that line. Let $$B$$ be any point on the plane different from $$A$$, and $$l'$$ the line obtained from connecting $$A$$ to $$B$$.

The case where $$l'$$ is parallel to $$l$$ is when $$B$$ is on $$l$$. How would you describe $$l'$$ relatively to $$l$$ in that case? Do you say they're equal, that they're the same line? Is there a dedicated word for that? Or does that make no sense?

Feel free to correct the wording, I'm not used to writing math in English.

Since we were already discussing this in the question you linked to (on french.stackexchange), I will post what I have been saying into this question, so that it may be reviewed by others.

This is Wikipedia's definition of "coincident points":

In geometry, two points are called coincident when they are actually the same point as each other.

By extension, I think it is correct to define two "coincident lines" by saying that two lines are called coincident when they are actually the same line as each other. Therefore, there is, in actuality, only a single line.

## Proof:

Assuming a Euclidean geometry, this Wikipedia article defines a line in terms of affine coordinates as follows:

$$L = \{ (x_1, x_2, \dots, x_n) | a_1 x_1 + a_2 x_2 + \dots + a_n x_n = c \}$$

Define two lines $$L_1, L_2 \in \mathbb{R}^n$$ as follows:

\begin{align} &L_1 = \{ (x_{1_1}, x_{1_2}, \dots, x_{1_n}) | a_{1_1} x_{1_1} + a_{1_2} x_{1_2} + \dots + a_{1_n} x_{1_n} = c_1 \} \\ &L_2 = \{ (x_{2_1}, x_{2_2}, \dots, x_{2_n}) | a_{2_1} x_{2_1} + a_{2_2} x_{2_2} + \dots + a_{2_n} x_{2_n} = c_2 \} \end{align}

Now, given our definitions of coincident points and lines, let's use the following definition:

Two lines are equivalent iff they are coincident.

And so, since we know that two lines are coincident iff all of their points are coincident, we have that

two lines are equivalent iff all of their points are coincident.

We are already assuming that $$L_1$$ and $$L_2$$ are coincident. Therefore, we only need to prove that, if $$L_1$$ and $$L_2$$ are coincident, then $$L_1$$ and $$L_2$$ are equivalent.

Since $$L_1$$ and $$L_2$$ are coincident, we have that

$$x_{1_i} = x_{2_j} \ \forall \ i = j \in \mathbb{N}$$

And so we have that

\begin{align} &(x_{1_1}, x_{1_2}, \dots, x_{1_n}) = (x_{2_1}, x_{2_2}, \dots, x_{2_n}) \in L_2 \Rightarrow L_1 \subset L_2 \\ &(x_{2_1}, x_{2_2}, \dots, x_{2_n}) = (x_{1_1}, x_{1_2}, \dots, x_{1_n}) \in L_1 \Rightarrow L_2 \subset L_1 \end{align}

Therefore, $$L_1 = L_2$$. $$\tag*{\blacksquare}$$

## EDIT:

After further research, it seems that the idea of whether a line is "parallel to itself" is a matter of choice, and will therefore depend on the definition given by the author.

According to the axioms of Euclidean geometry, a line is not parallel to itself, since it intersects itself infinitely often. However, some authors allow a line to be parallel to itself, so that "is parallel to" forms an equivalence relation.

This is Wikipedia's definition of parallel:

In geometry, parallel lines are lines in a plane which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel. Colloquially, curves that do not touch each other or intersect and keep a fixed minimum distance are said to be parallel.

Depending on how you define "touch", you could have a minimum distance of $$0$$ and still claim that the lines do not "touch".

And in this answer to a question asking whether a line is parallel to itself, Carl Mummert agrees with the Brilliant.org assessment, stating that it is a matter of opinion and the author's definition.

After reading all of this, I've come to the conclusion that I actually agree with the idea of a line being parallel to itself, since this is necessary for the relation "parallel" to qualify as an equivalence relation. This doesn't necessarily mean that Euclid's axioms are incorrect, but rather that we need to use a different, more modern, updated definition, so that it is consistent with current mathematics, as is discussed in the comments to the answers of the question that I linked above.

But does a line being parallel to itself imply that two coincident lines are actually two separate lines, rather than a single line? I can see how the English(/natural) language description implies this, since, as I just did, it begins by describing the existence of two coincident lines; that is, it begins by describing the existence of two objects. But as stated by Carl Mummert in his answer to the question that I linked to above, when we describe "two objects" in mathematics (in this case, two lines), we leave open the possibility that the two objects are actually equivalent (that is, that they are actually the same object). And since the language of mathematics is precise and rigorous, and natural language is certainly not, I think that the problem lies in the way the question is phrased in natural language: Asking "are two coincident lines parallel to themselves" is too ambiguous to be objectively translated into the language of mathematics. Instead, if one wants to ask such a question, they must phrase it in terms of mathematics, such as asking "are two coincident lines, where coincident is defined as (insert mathematical definition) and line is defined as (insert mathematical definition), parallel to themselves, where parallel is defined as (insert mathematical definition)?" I think that if you phrase it in this way, and if we accept as an axiom that equivalent objects are actually the same, single, unique object, then we inevitably come to the conclusion that there is only a single, unique object. So it seems to me that this all depends on whether we accept the axiom "equivalent objects are actually the same, single, unique object". Whether this is an axiom that should be (or is currently) accepted, based on our current understanding of mathematics, is something that I don't have enough understanding to comment on, so I will leave that for the more experienced people.

• I disagreed with your last point but it was way too off-topic on French SE to keep discussing it there. No one will disagree that $L_1 = L_2$ but you can still see them as two coincident lines. How is "For what interests us, two straight lines which never meet (or are coincident) are said to be parallel." contradictory? Parallel lines are lines that either share 0 point or share an infinity of points, that's what that definition mean. And even from your point of view, a line should be parallel to itself. – Teleporting Goat Jan 10 '20 at 16:02
• @TeleportingGoat This article brilliant.org/wiki/parallel-lines says the following: "According to the axioms of Euclidean geometry, a line is not parallel to itself, since it intersects itself infinitely often. However, some authors allow a line to be parallel to itself, so that "is parallel to" forms an equivalence relation." [...] – The Pointer Jan 10 '20 at 21:25
• @TeleportingGoat [...] Here is Wikipedia's definition of parallel: "In geometry, parallel lines are lines in a plane which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel. Colloquially, curves that do not touch each other or intersect and keep a fixed minimum distance are said to be parallel." Depending on how you define "touch", you could have a minimum distance of $0$ and still claim that the lines do not touch. [...] – The Pointer Jan 10 '20 at 21:26
• @TeleportingGoat [...] And in this math.stackexchange.com/a/45522/356308 answer to a question asking whether a line is parallel to itself, Carl Mummert agrees with the Brilliant.org assessment, stating that it is a matter of opinion. – The Pointer Jan 10 '20 at 21:26