I want to determine the Darboux reference for a given regular parametrization. I know the matricial expression for the natural parametrization but i was wondering whether, as it happens with Frenets reference, there was a proportionality constnat between the expression for a given parametrization and the natural one.
I've tried to determine this constant and, for the relationship of $t'=k_n N + k_g S$ I've got that, same as in frenet, for a given parametrization $t' = ||\gamma '(t)|| [k_n N ×+ k_g S]$.
However, I've tried to check whether this constant factor holds for all three of the derivatives of the frenet reference but I cant get it.
I'd apreciate if anyone could help me check if the factor holds.
Thanks in advance