Question Summary
I have a 2D parametric curve defined by two functions, $f_x(t)$ and $f_y(t)$, and by several parameters that adjust the overall shape of the curve. As-is, if $t$ varies at a constant speed, the velocity of the point $(f_x(t), f_y(t))$ follows a sine curve. I want to adjust the parametric functions so the velocity is constant, without changing the shape of the velocity curve. Looking for resources online, it sounds like this is sometimes possible with closed-form expressions and sometimes not. How do I determine if it is possible for this specific curve?
Equations and Parameters
Primary Curve: $$ f_x(t) = \cos(2 Q t) \frac{W}{2} + \left(\cos(2 Q t) \sin(Q N t) - \sin(2 Q t) \cos(Q N t) n - \sin\left(\frac{0 n}{2} - 2 Q D t\right)\right) \frac{S}{4 N D} \\ f_y(t) = \sin(2 Q t) \frac{W}{2} + \left(\sin(2 Q t) \sin(Q N t) + \cos(2 Q t) \cos(Q N t) n - \sin\left(\frac{π n}{2} - 2 Q D t\right)\right) \frac{S}{4 N D} \\ Q = \frac{π N}{π N W - 2 (N - D) S} \\ N = n - \sin\left(\frac{π}{2} n\right) \\ D = \frac{n + \sin\left(\frac{π}{2} n\right)}{2} \\ W = S + 2 A $$ where $n$ is an odd integer greater than 2, $S$ is a positive real number, and $A$ is a non-negative real number.
Velocity Along Curve: $$ v(t) = \sqrt{f_x'(t)^2 + f_y'(t)^2} = \bigl(N W - (N - D) S \sin(Q N x)\bigr) \frac{Q}{N} $$