loxodrome

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

Let

$\{\begin{array}{cc}x=r\sin u\cos v\hfill & \\ y=r\sin u\sin v\hfill & \\ z=r\cos u\hfill & \end{array}$

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

$bv=\ln \tan {\displaystyle \frac{u}{2}}+c,$ (1)

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

We denote  ${\displaystyle \frac{1}{b}}:=a$,  whence the equation of the family (1) in the parameter plane reads

$v=a\ln \tan {\displaystyle \frac{u}{2}}+ac:=v(u).$ (2)

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

$$\overrightarrow{k}=\overrightarrow{k}(u,v):=(r\sin u\cos v,r\sin u\sin v,r\cos u),$$

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

$$\overrightarrow{t}:=\frac{d}{du}\overrightarrow{k}(u,v(u))={\overrightarrow{k}}_u^{\prime }+{\overrightarrow{k}}_v^{\prime }\cdot v^{\prime }(u).$$

Since

$${\overrightarrow{k}}_u^{\prime }=(r\cos u\cos v,r\cos u\sin v,-r\sin u),{\overrightarrow{k}}_v^{\prime }=(-r\sin u\sin v,r\sin u\cos v,\mathrm{\hspace{0.17em}0})$$

and since

$$v^{\prime }(u)=\frac{a}{\tan \frac{u}{2}}\cdot \frac{1}{{\cos }^2\frac{u}{2}}\cdot \frac{1}{2}=\frac{a}{2\sin \frac{u}{2}\cos \frac{u}{2}}=\frac{a}{\sin u},$$

we can write the tangent vector of the curve as

$$\overrightarrow{t}=r\cdot (\cos u\cos v-a\sin v,\cos u\sin v+a\cos v,-\sin u).$$

For a tangent vector of a meridian, the partial derivative ${\overrightarrow{k}}_u^{\prime }$ may be taken.  Thus we obtain the value

$$\cos (\overrightarrow{t},{\overrightarrow{k}}_u^{\prime })=\frac{\overrightarrow{t}\cdot {\overrightarrow{k}}_u^{\prime }}{\left|\overrightarrow{t}\right||{\overrightarrow{k}}_u^{\prime }|}=\frac{1}{\sqrt{1+a^2}}=\frac{b}{\sqrt{b^2+1}},$$

which is a constant.  It means that the angle $\alpha$ 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