# Finding a mathematical representation of a string ball

I am looking for an elegant way is to describe the following mathematical model:

I want to create a random string-ball given only the length of the string and the diameter of the ball. Start point and endpoint do not matter that much. It does not have to be a closed-loop. Like one made out of wool which cats like to play a lot. Actually, I do not need a function. Individual equally spaced points are what I need in the end. However, having a function and then picking evenly spaced points seems the logical way to go for me. I have added an image to illustrate better what I'd like to create. The image shows an LED-band/string which is contained inside a ball.

I do not see which mathematical branch describes best what I am looking for. Probably it is related to mathematical functions, but it might be a topic of mathematical topologies, of which I do not have any prior knowledge.

My approach would be some iterative scheme. Choosing a start point, going in direction $$x$$ for a specific length. Then repeating this under the constraint of:

• staying inside the borders (ball)
• the first derivative before and after the previous point has to be the same
• the second derivate at the previous point has to be limited to prevent sharp bending.

I already see significant flaws in my approach, and it is not very elegant. A second approach which came to my mind is doing somewhat the opposite. First sampling random points inside the ball and then connecting them with a spline. To problem is that splines need the point in the correct order. Some heuristic might find a way to connect the points. This leads to something like a travelling salesman problem. I could imagine this might work. In my opinion, this is again a rather ugly solution.

Has there something similar already been done? I am happy if someone could point me in the right direction or help me to reformulate my problem into a domain that covers this.

• I am sitll not quite sure what you're looking for. Are you looking for uniformly sampled points on a ball (sphere + interior)? Jan 13, 2020 at 14:21
• No what I want is something that describes a string inside a ball. On this string are equally spaced points (e.g. the LEDs). It could for example be a function that creates the random string under certain constrains. For example, the string may not be discontinuous or has sharp edges. The equally spaced objects can then be found rather quick if such a function exists. Jan 13, 2020 at 14:38