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Loxodrome

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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

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

Comments

References

[a1] K. Strubecker, "Differentialgeometrie" , 2. Theorie der Flächenmetrik , de Gruyter (1958)
How to Cite This Entry:
Loxodrome. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Loxodrome&oldid=47718
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article