I'm trying to find equally spaced points on a parabola simply defined by $$y = \frac{x^2}{2 p}$$
Someone told me there is an easy way to split the parabola but he didn't tell me how and I cannot find it.
At first, I was trying to solve:
$$\begin{cases} \Delta x^2+\Delta y^2 = c^2\\ y+\Delta y = \frac{(x+\Delta x)^2}{2 p} \end{cases}$$
which lead to $$ \Delta x^4 + 4 x \Delta x^3 + (4x^2 + 4p^2)\Delta x^2 - 4p^2c^2 = 0 $$
which is not as simple as I fought it would be...
Actually, those are the equations if you want to find equally space point in 2D space but it will not split the parabola in equal length portions of parabola.
If splitting the parabola into equal length segments results in a simplier solution, I would want to know how to do it
The length of a segment of parabola over the interval $[a,b]$ is given by $$ \int_a^b\sqrt{1+y'^2}\,dx=\frac1p\int_a^b\sqrt{x^2+p^2}\,dx=\frac{x}{2}\,\sqrt{p^2+x^2}+\frac{p^2}{2}\,\log \bigl(\sqrt{p^2+x^2}+x\bigr)\Biggl|_a^b. $$ I think it is too complicated to be useful.