# loxodrome

Think in $\mathbb{R}^{3}$ 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

 $\displaystyle\begin{cases}x\;=\;r\sin{u}\cos{v}\\ y\;=\;r\sin{u}\sin{v}\\ z\;=\;r\cos{u}\end{cases}$

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

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

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

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

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

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

 $\vec{t}\;:=\;\frac{d}{du}\vec{k}(u,\,v(u))\;=\;\vec{k}^{\prime}_{u}+\vec{k}^{% \prime}_{v}\!\cdot\!v^{\prime}(u).$

Since

 $\vec{k}^{\prime}_{u}\;=\;(r\cos{u}\cos{v},\,r\cos{u}\sin{v},\,-r\sin{u}),\quad% \vec{k}^{\prime}_{v}\;=\;(-r\sin{u}\sin{v},\,r\sin{u}\cos{v},\,0)$

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

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

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

 $\cos(\vec{t},\,\vec{k}^{\prime}_{u})\;=\;\frac{\vec{t}\cdot\vec{k}^{\prime}_{u% }}{\left|\vec{t}\right||\vec{k}^{\prime}_{u}|}\;=\;\frac{1}{\sqrt{1\!+\!a^{2}}% }\;=\;\frac{b}{\sqrt{b^{2}\!+\!1}},$

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 Loxodrome 2013-03-22 19:11:02 2013-03-22 19:11:02 pahio (2872) pahio (2872) 11 pahio (2872) Definition msc 53A05 msc 53A04 msc 26B05 msc 26A24 meridian