Given the helicoid $$ \boldsymbol{r} = (u\sin v, u\cos v,v)$$ in three-dimensional Euclidean space, consider the triangle $T$ defined by $$ 0 \leq u \leq \sinh v, \qquad 0 \leq v \leq v_0.$$
The angles of the triangle are $\theta_1=\pi/4=\theta_2,\ \theta_3=\pi/2.$
Question: is the triangle geodesic? For a geodesic triangle, we have the equality (http://en.wikipedia.org/wiki/Gaussian_curvature#Total_curvature) $$ \sum_{i=1}^3 \theta_i - \pi = \int_T K \ \mathrm{d}A,$$ and given that the Gaussian curvature is $K=-1/(u^2+1)^2$ using the above data we get $$0 = \log\cosh v_0.$$ So the answer would seem to be NO: is that OK ?