Namespaces
Variants
Actions

Difference between revisions of "Loxodrome"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
(OldImage template added)
 
Line 51: Line 51:
 
$$
 
$$
  
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Strubecker,  "Differentialgeometrie" , '''2. Theorie der Flächenmetrik''' , de Gruyter  (1958)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Strubecker,  "Differentialgeometrie" , '''2. Theorie der Flächenmetrik''' , de Gruyter  (1958)</TD></TR>
 +
</table>
 +
 
 +
{{OldImage}}

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)


🛠️ This page contains images that should be replaced by better images in the SVG file format. 🛠️
How to Cite This Entry:
Loxodrome. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Loxodrome&oldid=53943
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article