# # of intersections for 2 curves on a plane

## Main Question or Discussion Point

What is the average number of intersections for two infinite curves confined to a plane?

Hootenanny
Staff Emeritus
Gold Member
What is the average number of intersections for two infinite curves confined to a plane?
How long is a piece of string?

Seriously, one can think of an infinite number of curves that intersect an infinite number of times (such as say y=sin(x) and y = 0). Equally, one can think of an infinite number of curves that intersect once (y = x and y =-x) or not at all (y=0 and y=1).

Perhaps I am missing the point of the question, but to me it doesn't make sense.

Hootenanny,

You gave three examples counting intersections for two infinite curves confined to a plane. I am asking what the overall average number of intersections is for all examples of infinite, coplanar, paired curves.

It may turn out that infinite sets cannot be averaged or proportioned, but if they can, I believe one of your answers is the correct number.

I am assuming that you do not necessarily mean “graph of a function” when saying “curve”. So a curve can cross itself. So an easier question to ask is “What is the average number of times a curve intersects itself?” This sounds like a random walk problem.
So, one could try to answer the following.
What is the probability that a random curve, starting at an initial point (wlog, (0,0)) intersects itself after traveling a distance (arc length) of d?

How do you define a random curve? A random walk is discrete, but random curve?

Last edited:
CRGreathouse
Homework Helper
How do you define a random curve? A random walk is discrete, but random curve?
Is the limit of a random walk as the step size goes to zero well-defined?

I only said that it sounds like a random walk problem. That is, it has that same flavor. I am sorry to have mentioned random walks. I was primarily trying to simplify the original question to “what is the probability that a curve intersects itself?” In fact, I am still uncertain what the original question is asking when using the phrase “infinite curve”. For, there are curves of infinite arclength contained in any bounded region of the plane. I cannot tell if the original question was about continuous functions for R to R, or about general infinite curves.

As far as defining a random curve in R^2:
Let C be the set of all curves in R^2 then let z be an arbitrary element of C. z is a random curve.
This is suitable for my proposed question:
“What is the probability that a random curve intersects itself within an arclength d?”
Of course you will need to restrict your overall set to the set of all curves in R^2 that have a well-defined arclength. Perhaps use the set of all smooth curves.

No, when you just "choose" an element out of the set of all curves it isn't "random" in the probabilistic sense. First of all can we even define a probability measure on that set? How big is it? It might be bigger than the set of reals.

The problem with the step size approach is that the limit might not be a curve, so there's no sense of talking about "intersecting" itself. For example, a fractal.

No, when you just "choose" an element out of the set of all curves it isn't "random" in the probabilistic sense. First of all can we even define a probability measure on that set? How big is it? It might be bigger than the set of reals.
Choosing an arbitrary element from a set is a random process.

Choose an element from the set {1,2,3,4,5}. What is the probability that the element is the number 4? 1/5. My “choice” is random in the probabilistic sense. If it was not, I cannot conclude the probability of choosing 4 is 1/5.

The set of all curves in R^2 is probably larger than the set of all reals. And the upper bound is the size of the set of all functions from R to R.

You can try making equivalence classes on the curves of length d. For example, two curves of length d are equivalent iff they share the same initial point and end point. This reduces the problem to subsets of R^2. But, in doing so, is there a distortion of the probabilities? Maybe an additional weight should be added to each class. Perhaps looking at the area of the region that is the union of all curves in one class. The union of all points on all curves in one class is an ellipse (except the degenerate case when the two end points are at a distance d.) There may be other options. How can you so quickly dismiss this direction? I am not going to write a paper on this problem. I am merely suggesting a technique to investigate the original question of the post. It is a difficult problem. My intuition is that the probability is 0. But, if you really want to know, give it a shot.

No, it's not random until you define a probability measure on it, then define a distribution by which you are picking the elements.

By your logic, the probability of choosing an element from $\aleph_7$ is [itex]1/\aleph_7[/tex].

I think this is a question that only the OP can answer.

Last edited:
It may turn out that infinite sets cannot be averaged or proportioned, but if they can, I believe one of your answers is the correct number.
Two infinite curves may have 0 to infinity intersections. The average of any (finite or infinite) set of numbers that are non-negative and contains at least 1 infinity is equal to infinity. Thus the answer to the original question is, statistically and definitively, infinity.

CRGreathouse
Homework Helper
No, when you just "choose" an element out of the set of all curves it isn't "random" in the probabilistic sense. First of all can we even define a probability measure on that set? How big is it? It might be bigger than the set of reals.
There are beth_2 functions from R to R, so their set is larger than the set of reals. But the set of smooth curves from R to R is only as large as the set of reals (beth_1).

The problem with the step size approach is that the limit might not be a curve, so there's no sense of talking about "intersecting" itself.
True, it might be a fractal. But I think a reasonable definition of intersection would be to consider curves to intersect if for some epsilon > 0 the random walks of step size epsilon intersect. (I can make this precise if desired, but there's nothing exciting about it.)

The set of all curves in R^2 is probably larger than the set of all reals. And the upper bound is the size of the set of all functions from R to R.
Yes, |R^R| is bigger than |R|. I'm pretty sure the curves on R^2 are upper-bounded by |R^R| = beth_2 -- just define them parametrically.

The average of any (finite or infinite) set of numbers that are non-negative and contains at least 1 infinity is equal to infinity.
Nah, I don't think that's reasonable. I'd much sooner disregard sets of measure 0 than have them override the rest of the set if it came to that.

But for this particular problem I can't see the answer being finite, so I agree with you there.

I'm sort of thinking that this question doesn't really have an answer.

I can ask what the average "real number" is, and it's all in how you look at it.

Can you even say what the average number of points of intersection is for the set of all lines in R^2? What if you restrict the lines to being in slope-intercept form having integer coefficients? That seems much more reasonable. I have a feeling somebody with some free time and a knack for linear algebra could put a number to that in short order.

Last edited:
junglebeast gives a convincing argument.

I had thought that because two infinite curves in one dimension (a straight line) always share an infinity of points, and in three dimensions seemingly on average zero points, that in two dimensions they would share one point ("average" between one and three dimensional case). Mine is primarily a qualitative (and not necessarily correct) observation.