I would like to plot the curve $\alpha(s) = (l(s), h(s))$, where $l$ and $h$ are the solutions to the system $l'^2 + h'^2 = 1$ and $l'h''-l''h'=h'\tan(l)$. Here's what I tried:
Clear[eq1, eq2, initConds, sol];
eq1 = l'[s] h''[s] - l''[s] h'[s] == h'[s] Tan[l[s]]
eq2 = l'[s]^2 + h'[s]^2 == 1;
initConds = {l[0] == 1, l'[0] == 1, h[0] == 0};
sol = NDSolve[{eq1, eq2, initConds}, {l, h}, {s, -10, 10}]
ParametricPlot[Evaluate[{l[s], h[s]} /. First[sol]], {s, -10, 10}]
This doesn't work (I don't get any image). Why?
Table[Evaluate[{l[s],h[s]}/.First[sol]], {s,-10,10,1/10}]
might give you some idea what is happening in this case and in other cases in the future. This can let you realize things like there are complex numbers for some or all of the graph or there are undefined variables or other problems that plot might not clearly explain to you. $\endgroup$