Difference between revisions of "Loxodrome"
From Encyclopedia of Mathematics
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| − | A curve on a [[Surface of revolution|surface of revolution]] that cuts all the meridians at a constant angle | + | <!-- |
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| + | $#C+1 = 6 : ~/encyclopedia/old_files/data/L060/L.0600970 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" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l060970a.gif" /> | ||
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For surfaces of revolution whose [[First fundamental form|first fundamental form]] can be written as | 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 | 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 | 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 | 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==== |
| + | <table> | ||
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Strubecker, "Differentialgeometrie" , '''2. Theorie der Flächenmetrik''' , de Gruyter (1958)</TD></TR> | ||
| + | </table> | ||
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Latest revision as of 18:40, 13 May 2023
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) |
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How to Cite This Entry:
Loxodrome. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Loxodrome&oldid=18141
Loxodrome. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Loxodrome&oldid=18141
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article