The above is an example of a “wild arc”: it is an arc in space (a continuous image of the unit interval ) that is so pathologically embedded that it cannot be “straightened out” by a deformation of 3-space. Or if you have had some calculus, it is impossible to define a tangent vector to the arc at two points; in this case, the end points.
Now do you see the red circles around one of the end points? Those represent the equator of a round sphere whose “center” is at the end point. As you can see, the arc hits those spheres at 3 points. So, take any old sphere that has that endpoint “inside of it” (technically, inside the ball bounded by the sphere, and for you experts, we insist that the sphere be a “smooth” sphere) . Now, what is the fewest number of points that this arc meets such a sphere? That is called the “penetration index” of the endpoint.
It sure looks like the penetration index is 3, but all this picture does is to show that the penetration index is at most three. How do you know that there isn’t some sphere that you haven’t thought of that contains the endpoint inside of it (ok, inside of the region bounded by the sphere) that hits this arc in two points, or even one point?
It turns out that no such sphere exists, but that requires proof and the proofs aren’t always that easy.
I bring this up because I was doing a calculation similar to this and wondered why it was getting so complex; then I realized that my calculation was akin to calculating a “penetration index”.