# Division of mathematically defined curves using arc length formula. Examples in Desmos Graphing Calculator. Link attached.

I am trying to divide some simple curves into equal segments. I managed to use arc length formula but then I got stuck. What to do next to divide any of these curves into let's say 10 equal segments?

I provided a link containing 4 examples that I would like to figure out: https://www.desmos.com/calculator/ks9o0p59tg

Having this solved in Desmos would be of great help. Desmos is a free online platform if you want to give a try and solve any of the provided examples. Thank you!

Construct a parameterization of the curve $$f(x)$$ with respect to arc length. Since you did not do this, I assume you are unfamiliar with parameterization. This is a mathematical procedure in which we express the coordinates of each point on the curve as some function of a common parameter, say, $$t$$. As an example, the unit circle has a parameterization $$x=cos(t)$$, $$y=sin(t)$$ for $$0\leq t\leq 2\pi$$. Notice that in general parameterizations are not unique, i.e. we could let $$2\pi\leq t\leq 4\pi$$. The goal, then is to construct a parameterization where a 'step' in $$t$$ corresponds to a 'step' in arc length. You can use the arc length formula to find the arc length of the curve in terms of your parameterization variable $$t$$, and the formula I assume you're using is a special case of this more general form: $$s(t)=\int_0^t\sqrt{x'(u)^2+y'(u)^2}du$$ where $$u$$ is some dummy variable substitution $$u=t$$. This will spit out some function of $$t$$, and solving $$s=f(t)$$ for $$t$$ in terms of $$s$$ will give a function you plug back in to your parameterization to make the parameter (now $$s$$) exactly equal to arc length over some new interval. Divide this interval up into however many pieces, and then plug the value of the n-th boundary into your paraneterizations to get values of $$x$$ and $$y$$. Once I get off mobile I'll see if I can cook up some Desmos thing.