I would like to plot the curves given by $\alpha(s) = (u(s), v(s))$, where $u$ and $v$ are like below.
Clear[u, v];
{u, v} = {u[t], v[t]} /.
NDSolve[{u'[t]^2 + v'[t]^2 == 1,
u'[t] v''[t] - u''[t] v'[t] == u[t] v'[t] - v[t] u'[t],
u'[0] == Sin[0.18], v'[0] == Cos[0.18]}, {u[t], v[t]}, {t, -7.5,
7.5}][[1]];
What I want to do is, for each different initial conditions $u'(0) = \cos(\theta)$, $v'(0) = \sin(\theta)$, $u'(t)v''(t) - u''(t)v'(t) = -c(u'(t)v(t) - u(t)v'(t))$ with $\theta$ varying in $[0, 2\pi]$ and $c$ varying in, say, $(0, 10]$, plot a new curve. I'm new to Mathematica so I still have a hard time doing these little things.