Notation for point of intersection in euclidean geometry

Question

Since there are signs for things such as paralell, $\parallel$, and perpendicular, $\perp$, is there any sign for a point of intersection? Should one use $\cap$? This would be useful in writing instructions for constructing some geometric shapes.

Answer 1

Short answer: there is no well-known notation that doesn't overlap with a different usage in a different area of mathematics, so best define the symbols you want to use before first usage.

When you treat a line as a set of points, then $g\cap h=\{P\}$ would be accepted set-theoretic notation for that as intersection of sets. People will likely understand $P=g\cap h$ without curly braces to mean the same, particularly since "$\cap$" is pronounced as "intersect" in set context which matches your geometric meaning, too. Strictly speaking it's a different use of that symbol, though.

In many areas a line is not just a set of points, but a geometric entity in its own regard. In the lectures I heard (and later taught) we used $\wedge$ to denote intersection, and pronounced the operation "meet". The counterpart to this was $\vee$ pronounced "join" to denote the operation of joining two points to obtain a line through them. All of this was in a context of projective geometry, where the roles of points and lines are fairly symmetric, so having similar symbols for these fundamental operations was very useful.

This use of $\wedge$ was I believe inspired by the set notation, but the symbol is also used in logic to denote "and". It would be a poor choice in situations where both logic and geometry are used. And it does require a definition at first use, to avoid confusing the audience.

The symbols and operator names for join and meet are also described in a Wikipedia article in the context of ordered sets and lattices. There is a relation to the geometric operations I described but it's not as obvious as an explicit definition in a given context would make it.

Yet another usage of $\wedge$ is in exterior algebra but the conventional geometric interpretation of that is more like the "join" I defined above. I guess by the time people are familiar with exotic things like exterior algebra, they are also familiar with symbols changing their meaning depending on context.

Using $+$ for this operation, as you suggest in a comment, would feel weird to me. But that may be because I tend to perform many of my geometric operations at an algebraic level using coordinates, so I would be inclined to read that as vector addition. If you are speaking to an audience not familiar with coordinate-based geometry, using plus might be an option.