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