View source for Loxodrome

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A curve on a [[Surface of revolution|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.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l060970a.gif" />

Figure: l060970a

For surfaces of revolution whose [[First fundamental form|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 ) .
$$

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Strubecker,   "Differentialgeometrie" , '''2. Theorie der Flächenmetrik''' , de Gruyter  (1958)</TD></TR></table>