Did you make the animation or you found it somewhere in the internet?
Do you have a formula for each of these things?
What do you mean by "constant time"? From your description it seams all the points have a closed formula, there are no iterative steps that may be long? Do you want a simpler direct formula?
the animation is something i constructed in order to help facilitate discussion and get help :)
i've outlined the computations more detail in a reply below
by constant time i mean O(1)
computing the white dot currently requires me to sweep the already computed loop points which isn't a big deal in a lot of use cases, but with audio its a show stopper
the white dot is also the key to a much more complex computation that im using
Do you have a formula for each of these things?
What do you mean by "constant time"? From your description it seams all the points have a closed formula, there are no iterative steps that may be long? Do you want a simpler direct formula?