Loxodrome

A curve on a surface of revolution that cuts all the meridians at a constant angle $ \alpha $. If $ \alpha $ is an acute or obtuse angle, then a loxodrome performs infinitely many windings about the pole, getting closer and closer to it.

Figure: l060970a

For surfaces of revolution whose first fundamental form can be written as

$$ ds ^ {2} = du ^ {2} + G ( u) d v ^ {2} , $$

the equation of a loxodrome is

$$ v \mathop{\rm cotan} \alpha = \pm \int\limits _ { u _ 0 } ^ { u } \frac{du }{ \sqrt {G ( u) } } . $$

For a sphere with first fundamental form

$$ d s ^ {2} = R ^ {2} ( d u ^ {2} + \cos ^ {2} u d v ^ {2} ) $$

the equation of a loxodrome is

$$ v \mathop{\rm cotan} \alpha = R \mathop{\rm ln} \mathop{\rm tan} \left ( \frac \pi {4} + \frac{u}{2R} \right ) . $$

References

[a1]	K. Strubecker, "Differentialgeometrie" , 2. Theorie der Flächenmetrik , de Gruyter (1958)