A loxodrome is a curve on the unit sphere that intersects the meridians at a fixed angle, say $\alpha$. Show that, in the Mercator surface patch $\pmb\sigma(u,v)=(sech(u)cos(v),sech(u)sin(v),tanh(u))$, a unit-speed loxodrome satisfies $\dot u=cos(\alpha)cosh(u)$, $\dot v=\pm sin(\alpha)cosh(u)$. (a dot denoting differentiation with respect to the parameter of the loxodrome). Deduce that loxodromes correspond under $\pmb\sigma$ to straight lines in the $uv$-plane.
Pressley has answers at the back, and he says $\pmb\sigma_u$ is tangent to the meridians. How is this so? Isn't $\pmb\sigma_u$ tangent to the parallels?