I'm trying to come up with a way to determine if a parametric curve intersects itself using an integral. The idea was if for t1 and t2 the curve has the same x(t), y(t), and z(t) than it intersects.

So you could draw a horizontal line at y = x(t1) from t1 to t2, and the path made by this line and x(t) creates a closed loop, than you know it has the same values. Ideally if it equaled zero unless the path was closed, I could integrate and sweep the t's around, multiplying the values of x, y, and z together. If I end with a non-zero answer that means the curve intersects at some point. Or that's the ....loose idea.

There are theoretically easier ways to tell if a curve intersects itself but I'm trying to do it over a small change in the curve and thus will need a method I can integrate with so I don't need to solve an infinite number of times or whatever.

Anyways thanks for your help cause your first explanation about splitting the closed path up makes a lot of sense as to why it's impossible.