# If two curves only touch, do they technically "intersect"?

I was playing sprouts with a few friends the other day, and one of them tried to be clever by "squeezing" their line next to another player's line to effectively prevent any passthrough play:

Now, I'm pretty terrible with the trackpad in paint, so bare with my low quality art here. The focus is the upper right area where the red and blue lines look like they're intersecting (just imagine they aren't as it's a side effect of my terrible paint skills). The idea the red player had was to shrink the playable region between their line and the blue player's line as much as possible, effectively closing the area off.

However, this annoyed both myself (green) and the red player. Since none of us could truly determine if the rules permitted this movement, we all agreed at the time that since the rules prevent intersection, the space between the red and blue lines was just really small. As such, pass through could still occur with the understanding that the line passing through would "shrink" into the space and expand back to normal size on the other side, effectively simulating a pass through without intersection.

Unfortunately though, my curiosity is getting the better of me and now I'm wondering if it's wrong to assume this since two objects must overlap to be considered an intersection, by definition:

(of two or more things) pass or lie across each other.

Or not? If the outermost edges of two objects only touch, is it technically an intersection?

Edit: For close votes related to this being off-topic due to not being about mathematics; my question isn't focused on the game of sprouts, but rather something I encountered during play that I was looking for clarification on. That something is terminology related to intersection and is mathematics related.

• What you're describing is called a "point of tangency," and I think the standard definition of "intersection" (which is basically a set-theoretic definition: an intersection of two curves is any point that lies on both of them) would include points of tangency. But for some problems, it might be more convenient to define "intersection" to include only crossing rather than tangent points - the answer to questions of the form "does thing fit the definition of word" is often "it depends which definition." Commented Oct 17, 2021 at 14:44
• I don't think that you will find the answer in mathematics. As Connor days: it depends on which definition. It might be surprising but there is no international body setting the rules of mathematics. They is no "official" definition. In some cases, pretty much everyone agrees but that's as close as it gets. You might have more luck in a games forum. If not then agree a house rule before playing again. Commented Oct 17, 2021 at 14:54
• For what it's worth, Wikipedia's "Sprouts (game)" entry provides the rule "The line [added by a player] may be straight or curved, but must not touch or cross itself or any other line."
– Blue
Commented Oct 17, 2021 at 15:37
• A topologist would say that curves $C_1, C_2$ in the plane "cross" if at the point of intersection, there are points of $C_1$ which lie on the opposite sides of $C_2$ (one can make this precise if you like). Otherwise, one says that the curves "touch" each other at the point of intersection. Commented Oct 17, 2021 at 23:22
• A technical word for (really) "intersecting lines" is "transversal lines" Commented Oct 18, 2021 at 16:18

## 3 Answers

In the game of sprouts, you draw "lines" between "spots". The lines do not have to be straight, so we might call them arcs or curves instead. But let's continue to call them lines, just realizing that we do not mean "straight line".

The question is, does a line in sprouts have a thickness?

The drawing of a line obviously does have a thickness, otherwise we would not be able to see it. You can even approximately measure the thickness of a line you draw. Since the drawing of a line has a thickness, it covers an area on the plane.

On the other hand, a mathematical curve between two points has no thickness and covers no area on the plane.

If the lines have a minimum thickness (as your drawings of them do), there is not enough space between the red and blue lines to draw another line. The green line must intersect the red or blue line (or both) because it will necessarily overlap one line or the other if you try to draw a green line through that space.

On the other hand, if we consider the lines as mathematical curves on a continuous plane, then the drawings you make are only approximations of the curves; the actual curves run somewhere down the middle of the drawings of them. In that interpretation, we can assert that the red and blue lines do not intersect (the actual lines pass somewhere close to, but apart from, the place where the drawings of them intersect). If two curves run alongside each other like this without intersecting, it is always possible to pass a third curve through the space between the two curves. Your drawing of the green line could be seen as a schematic of how the ideal curves actually will be arranged. Viewed that way, the move should be considered legal.

By the way, in that interpretation even the fact that the drawings of the red and blue lines touch would be considered an artifact of the drawing and not an actual intersection.

Since this game was invented by mathematicians, I would tend to think that they meant the "lines" to be mathematical curves rather than objects with a thickness and an area on the plane. Another argument for this viewpoint is that the game is apparently meant to be a game of logic rather than a game dependent on drawing skill. There are mathematical statements about the length of a game that would seem to depend on this interpretation; in particular, I think the formula for the minimum length is not true if you consider the lines to have thickness.

As a practical matter, good etiquette in the game would seem to require that when drawing a line, you leave enough space for additional lines to be drawn. The blue player should have drawn the line much further away from the spot, and the red line should have been drawn midway between the spot and the blue line.

A technical term for "just touch" is "tangency". In common usage of the word "intersect" and also the English Wikipedia page for "tangent", this would still count as an intersection. I also don't think this sort of move is allowed in Sprouts, just based on how the game has been analyzed.

A non-tangent intersection point would be a transverse intersection.

Here are two circles that touch (they have exactly one point in common). When seen as sets of points on the plane, they "intersect" in the sense that the set of common points is non-empty. However, they do not intersect in the sense of "crossing": The smaller circle is entirely inside the larger one (it could equally well be entirely outside). Straight lines cannot touch without intersecting in this stronger sense (except at the ends, but then they are technically "line segments"), but non-straight lines can.

Now about sprouts. If you take "intersect" to have the stronger sense, your version of the sprouts rules might forbid lines that intersect, or it might forbid lines that touch (except for the endpoints of course). But either way, Red cannot prevent Green from getting through:

• If lines may not touch, there will be a channel, however small, that the green line can go through.
• If lines are allowed to touch but not intersect, well, Green can draw a line that touches the red and green lines at the same point that they touch each other, and get past without intersecting with either one!