loxodrome

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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 poles. 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.

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