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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?

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I realized what my mistake was. In the Mercator projection, meridians correspond to the parameter curves $v=constant$. Therefore, in order to find the tangent vector for that curve we need $\pmb\sigma_u$, (since $\pmb\sigma_u$ assumes $v$ is a constant).

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