The curve looks good but I’m worried by the denominator. How do you make sure b x^2 + x + c \neq 0, \forall x \in \mathbb{R} or at least \forall x \in [0 ; \text{toe}] ?
Yeah but slope = 1 is not really a use case here, so I wouldn’t put too much emphasis on getting a straight line.
I’m starting to wonder if it wouldn’t be possible to keep current 3rd/4th orders and add constraints over the derivatives in the solver (like y''(x) \neq 0 to ensure mononicity and y'(x) > 0 to avoid gradient reversal). So that would yield an optimization more than a solving. Something like that : https://fr.mathworks.com/help/optim/ug/lsqlin.html. The reason is higher order splines should be able to model any “round things”, I don’t like having to divide by a polynomial (mind those zeros and their neighbourhood, that will cause arithmetic problem and float denormals that will slow the vector vode), and so far, the splines are handled uniformly by a vectorized routine of FMA that is very efficient. As much as possible, I would like to stick to polynomials.
EDIT : https://stanford.edu/class/engr108/lectures/constrained-least-squares_slides.pdf Just spot on.