Let c be a curve such that $R\rightarrow R^3$. Let c’(t) be the derivative of c with respect to some parameterization t.
If $c(t_o) = 0$ for some $t_o$, what is the significance? Is the existence of c'=0 independent of parametrization.
Context: Working on a numeric a numeric integration of bios-savarte law. Decided to plot abs of my tan vector to my curve and noticed it hit zero, so now I’m trying to diagnose the issue. Ie is it error in coding/plotting or am in for a big set back. I need the unit tan vector at each point for Bios-Savart.
The particular curve I am trying to describe is a toroidal coil. I found this equation, $$c(t,r,a,n) = (-r \sin(t/n) - a \cos(t) \sin(t/n),r \cos(t/n) + a \cos(t) \cos(t/n), a \sin(t))$$ by starting with a parametrized helix and good old fashion trial and error. The $a$ is the small radius. The $r$ is the big radius, and the $n$ is the number of loops.
Physically, it seems like $c'(t)$ should always be nonzero for a toroidal coil which makes me think I either don't understand $c'= 0$, or my function is wrong.
Turns out that the c’(t) outlined above never intersected zero. However the plot appeared to when n>>s>>a. It ended up being a non problem. Great fun to have that cleared up