Namespaces
Variants
Actions

Difference between revisions of "Loxodrome"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A curve on a [[Surface of revolution|surface of revolution]] that cuts all the meridians at a constant angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060970/l0609701.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060970/l0609702.png" /> is an acute or obtuse angle, then a loxodrome performs infinitely many windings about the pole, getting closer and closer to it.
+
<!--
 +
l0609701.png
 +
$#A+1 = 6 n = 0
 +
$#C+1 = 6 : ~/encyclopedia/old_files/data/L060/L.0600970 Loxodrome
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
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" />
Line 7: Line 21:
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060970/l0609703.png" /></td> </tr></table>
+
$$
 +
ds  ^ {2}  = du  ^ {2} + G ( u)  d v  ^ {2} ,
 +
$$
  
 
the equation of a loxodrome is
 
the equation of a loxodrome is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060970/l0609704.png" /></td> </tr></table>
+
$$
 +
v  \mathop{\rm cotan}  \alpha  = \pm  \int\limits _ { u _ 0 } ^ { u } 
 +
\frac{du }{
 +
\sqrt {G ( u) } }
 +
.
 +
$$
  
 
For a sphere with first fundamental form
 
For a sphere with first fundamental form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060970/l0609705.png" /></td> </tr></table>
+
$$
 +
d s  ^ {2}  = R  ^ {2} ( d u  ^ {2} + \cos  ^ {2}  u  d v  ^ {2} )
 +
$$
  
 
the equation of a loxodrome is
 
the equation of a loxodrome is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060970/l0609706.png" /></td> </tr></table>
+
$$
 
+
v  \mathop{\rm cotan}  \alpha  = R  \mathop{\rm ln}  \mathop{\rm tan} \left (
 +
\frac \pi {4}
  
 +
+
 +
\frac{u}{2R}
 +
\right ) .
 +
$$
  
 
====Comments====
 
====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>

Revision as of 04:11, 6 June 2020


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=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