Think in 3 a sphere with radius r and two antipodal points of it wich we call the North poleMathworldPlanetmath and the South pole.  MeridiansMathworldPlanetmath are great circles passing through the .  A loxodrome is a curve on the sphere intersecting all meridians at the same angle.



be a parametric presentation of the sphere (cf. the spherical coordinatesMathworldPlanetmath).  We will show that

bv=lntanu2+c, (1)

where b and c are constants, is an equation of loxodromes in the Gaussian coordinates u,v.

We denote  1b:=a,  whence the equation of the family (1) in the parameter plane reads

v=alntanu2+ac:=v(u). (2)

When we denote also the position vector of a point of the sphere by


we have the tangent vectorMathworldPlanetmath of a curve (1) on the sphere:



ku=(rcosucosv,rcosusinv,-rsinu),kv=(-rsinusinv,rsinucosv, 0)

and since


we can write the tangent vector of the curve as


For a tangent vector of a meridian, the partial derivativeMathworldPlanetmath ku may be taken.  Thus we obtain the value


which is a constant.  It means that the angle α between the curve (1) and a meridian is constant.

Pictures in http://hu.wikipedia.org/wiki/LoxodromaWiki

Title loxodrome
Canonical name Loxodrome
Date of creation 2013-03-22 19:11:02
Last modified on 2013-03-22 19:11:02
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Definition
Classification msc 53A05
Classification msc 53A04
Classification msc 26B05
Classification msc 26A24
Defines meridian