If it is the lengths of the straight lines between the points on the curve you want equal, I think the only way is to solve it iteratively:
(I was actually surprised this works - my first intuition was that it was an overconstrained problem and there wouldn’t generally be a solution. It reminds me a bit of the 4-legged table problem: https://arxiv.org/pdf/math-ph/0510065v6.pdf)