Loxodromic curve

A loxodromic curve (of the Greek grc lox (O) - and - grc dromy race obliques ) is a curve which cut the Méridien S under a constant angle .

A loxodromic road is represented on a Sea chart or aeronautics in Projection of Mercator by a straight line but does not represent the the shortest distance between two points. Indeed the shortest road is called orthodromic road or Orthodromie.

The loxodromic road is a road with constant course.

Loxodromic navigation

The problem arising is that of the determination of the road and the loxodromic distance between two points. It is thus about the opposite problem of the Navigation to the regard.

• if the two points has and B is not very distant, one can be satisfied with approximate formulas (average latitude):

$M\; \backslash ,$ being the distance covered with the road $R\_v\; \backslash ,$; $\backslash \; varphi\_A,\; G\_A\; \backslash ,$ and $\backslash \; varphi\_B,\; G\_B\; \backslash ,$les coordinated geographical (latitude, longitude) of the points has and B, and $\backslash \; varphi\_m\; =\; \backslash \; frac\; \{\backslash \; varphi\_A\; +\; \backslash \; varphi\_B\}\; \{2\}\; \backslash ,$:

$\backslash \; tan\; R\_v\; =\; \backslash \; frac\; \{G\_B\; -\; G\_A\}\; \{\backslash \; varphi\_B\; -\; \backslash \; varphi\_A\}\; \backslash \; cos\; \backslash \; varphi\_m\; \backslash ,$

and: $M\; =\; \backslash \; frac\; \{\backslash \; varphi\_B\; -\; \backslash \; varphi\_A\}\; \{\backslash \; cos\; R\_v\}\; \backslash ,$

these approximate formulas remain precise near with 1 nautical (1 ' of arc) for nautical.

• exact formulas ( increasing latitudes of the projection of Mercator):

$\backslash \; tan\; R\_v\; =\; -\; \backslash \; frac\; \{G\_B\; -\; G\_A\}\; \{\backslash \; lambda\_B\; -\; \backslash \; lambda\_A\}\; \backslash ,$

and: $M\; =\; \backslash \; frac\; \{\backslash \; varphi\_B\; -\; \backslash \; varphi\_A\}\; \{\backslash \; cos\; R\_v\}\; \backslash ,$

$\backslash \; lambda\; \backslash ,$, in 'arc, is called the increasing latitude (formerly given in the tables of Friocourt);

$\backslash \; lambda\; =\; 7915,7.\; \backslash \; ln\; \backslash \; tan\; (\backslash \; frac\; \{\backslash \; pi\}\; \{4\}\; +\; \backslash \; frac\; \{\backslash \; varphi\}\; \{2\})\; \backslash ,$

Mathematical demonstration

On the terrestrial sphere, the loxodromic curves correspond (when they “are not degenerated”, i.e. when the initial angle given is not null) to spirals being rolled up around the pole (the north pole if the initial angle and in $]\; 0,\; \backslash \; pi\; and\; that\; displacement\; is\; done\; in\; the\; direction\; of\; [[latitude]\; S\; increasing).$

Is to be determined an equation of the loxodromic curve and to calculate the length traversed with constant course $\backslash \; alpha\; \backslash \; in\; \backslash ,]\; 0,\; \backslash \; pi$

Let us consider the usual Coordonnée S spherical on the sphere unit: the Longitude $\backslash \; varphi$ and the colatitude $\backslash \; theta$. The loxodromic curve constitutes a arc on the sphere which one supposes of class $C^1$: $\backslash \; varphi\; \backslash \; mapsto\; \backslash \; theta\; (\backslash \; varphi)$; that is to say the function $f:\; \backslash \; varphi\; \backslash \; mapsto\; M\; (\backslash \; varphi,\; \backslash \; theta\; (\backslash \; varphi))$ which with longitude $\backslash \; varphi$ associates the point running of the loxodromic curve of longitude $\backslash \; varphi$ and of colatitude $\backslash \; theta\; (\backslash \; varphi)$. It is thus necessary well-sure to give oneself at the beginning an origin longitudes, since to a $\backslash \; varphi$ given to $2\; \backslash \; pi$ close, corresponds an infinity of distinct points on the arc, of different colatitudes. Let us leave the equator and follow the loxodromic curve towards the north pole by refusing us the classes modulo $2\; \backslash \; pi$ for $\backslash \; varphi$: for example .

A tangent vector with the loxodromic curve is thus $f\text{'}\; (\backslash \; varphi)\; =\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; (\backslash \; varphi,\; \backslash \; theta\; (\backslash \; varphi))\; +\; \backslash \; theta\text{'}\; (\backslash \; varphi)\; \backslash \; cdot\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; (\backslash \; varphi,\; \backslash \; theta\; (\backslash \; varphi))$. This vector, which directs the tangent to the arc, thus forms, by assumption, an angle $\backslash \; alpha$ with any vector (not no one) directing it parallel with the point considered. Vector directing it a parallel in $M\; (\backslash \; varphi,\; \backslash \; theta\; (\backslash \; varphi))$ is $\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; (\backslash \; varphi,\; \backslash \; theta\; (\backslash \; varphi))$ (while a vector directing the meridian line is well-sure $\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; (\backslash \; varphi,\; \backslash \; theta\; (\backslash \; varphi))$).

In the continuation, to reduce the writing, one will not clarify any more the point $(\backslash \; varphi,\; \backslash \; theta\; (\backslash \; varphi))$ to which are taken the partial functions and their derivative.

By carrying out the scalar product of a directing vector of the tangent to the loxodromic curve and of a directing vector of the parallel, one obtains the product of the standards of these vectors by the cosine of the angle which they form:

$\backslash \; left\; (\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash ;\; |\backslash ;\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; +\; \backslash \; theta\text{'}\; (\backslash \; varphi)\; \backslash \; cdot\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; \backslash \; right)\; =\; \backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash |\backslash ,\; \backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; +\; \backslash \; theta\text{'}\; (\backslash \; varphi)\; \backslash \; cdot\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; \backslash |\backslash \; cdot\; \backslash \; cos\; \backslash \; alpha$, by noting $(\backslash \; vec\; U\; \backslash ;\; |\backslash ;\; \backslash \; vec\; v)$ the scalar product $\backslash \; vec\; u$ by $\backslash \; vec\; v$.

While raising squared:

$\backslash \; left\; (\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash ;\; |\backslash ;\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; +\; \backslash \; theta\text{'}\; (\backslash \; varphi)\; \backslash \; cdot\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; \backslash \; right)\; ^2=\; \backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash |^2\; \backslash ,\; \backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; +\; \backslash \; theta\text{'}\; (\backslash \; varphi)\; \backslash \; cdot\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; \backslash |^2\backslash cdot\; \backslash cos^2\; \backslash alpha$.

One has in addition clearly: $\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash \; club-footed\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}$ (the parallels and the meridian lines are orthogonal). Therefore, by application of the theorem of Pythagore, the expression is reduced to:

$\backslash \; left\; (\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash ;\; |\backslash ;\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash \; right)\; ^2=\; \backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash |^2\; \backslash ,\; \backslash \; left\; (\backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash |^2\; +\; \backslash \; theta\text{'}^2\; (\backslash \; varphi)\; \backslash |\backslash \; cdot\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; \backslash |^2\; \backslash \; right)\; \backslash \; cdot\; \backslash \; cos^2\; \backslash \; alpha$.

And while simplifying:

$\backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash |^2=\; \backslash \; left\; (\backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash |^2\; +\; \backslash \; theta\text{'}\; (\backslash \; varphi)\; ^2\; \backslash \; cdot\; \backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; \backslash |^2\; \backslash \; right)\; \backslash \; cdot\; \backslash \; cos^2\; \backslash \; alpha$.

From where, with “$1-\; \backslash \; sin^2\; =\; \backslash \; cos^2$”

$\backslash \; sin^2\; \backslash \; alpha\; \backslash \; cdot\; \backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash |^2=\; \backslash \; theta\text{'}\; (\backslash \; varphi)\; ^2\; \backslash \; cdot\; \backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; \backslash |^2\; \backslash \; cdot\; \backslash \; cos^2\; \backslash \; alpha\; \backslash \; qquad\; \backslash \; mathbf\; \{(1)\}$.

Let us calculate the two standards intervening in this equation:

it is known, according to the spherical parameter setting brought back to the Cartesian coordinates in the base $(\backslash \; vec\; I,\; \backslash \; vec\; J,\; \backslash \; vec\; K)$, that $\backslash \; overrightarrow\; \{OM\}\; (\backslash \; varphi,\; \backslash \; theta)\; =\; \backslash \; cos\; \backslash \; theta\; \backslash ;\; \backslash \; vec\; K\; +\; \backslash \; sin\; \backslash \; theta\; \backslash ;\; \backslash \; vec\; u\_\; \backslash \; varphi$, where $\backslash \; vec\; u\_\; \backslash \; varphi$ is the radial unit vector of the equatorial plan defined by: $\backslash \; vec\; u\_\; \backslash \; varphi\; =\; \backslash \; cos\; \backslash \; varphi\; \backslash ;\; \backslash \; vec\; I\; +\; \backslash \; sin\; \backslash \; varphi\; \backslash ;\; \backslash \; vec\; j$. One defines $\backslash \; vec\; v\_\; \backslash \; varphi$ like the vector derived compared to $\backslash \; varphi$ from $\backslash \; vec\; u\_\; \backslash \; varphi$: $\backslash \; vec\; v\_\; \backslash \; varphi\; =\; \{D\; \backslash \; vec\; u\_\; \backslash \; varphi\; \backslash \; over\; D\; \backslash \; varphi\}\; =\; \backslash \; sin\; \backslash \; varphi\; \backslash ;\; \backslash \; vec\; I\; +\; \backslash \; cos\; \backslash \; varphi\; \backslash ;\; \backslash \; vec\; j$. Then $\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; varphi\}\; =\; \backslash \; sin\; \backslash \; theta\; \backslash ;\; \backslash \; vec\; v\_\; \backslash \; varphi$ and $\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; theta\}\; =\; -\; \backslash \; sin\; \backslash \; theta\; \backslash ;\; \backslash \; vec\; K\; +\; \backslash \; cos\; \backslash \; theta\; \backslash \; vec\; u\_\; \backslash \; varphi$. Thus, $\backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; \backslash |=\; \backslash \; sin\; \backslash \; theta$ and $\backslash |\{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; \backslash |=1$.

The equation $\backslash \; mathbf\; \{(1)\}$ is reduced to:

$\backslash \; sin^2\; \backslash \; alpha\; \backslash \; sin^2\; \backslash \; theta\; (\backslash \; varphi)\; =\; \backslash \; theta\text{'}^2\; (\backslash \; varphi)\; \backslash \; cos^2\; \backslash \; alpha$

and since one supposed a way towards the north pole, $\backslash \; theta$ is a decreasing function of $\backslash \; varphi$ and there , one supposes moreover $\backslash \; alpha\; \backslash \; in\; \backslash ,]\; 0,\; \backslash \; pi/2\; the\; other\; cases,\; one\; deduces\; the\; arc\; by\; a\; central\; symmetry\; and/or\; a\; suitable\; rotation\; (S),\; therefore\; one\; does\; not\; lose\; a\; general\; information),\; consequently:\; :$ \backslash \; sin\; \backslash \; alpha\; \backslash \; sin\; \backslash \; theta\; (\backslash \; varphi)\; =\; \backslash \; theta\text{'}\; (\backslash \; varphi)\; \backslash \; cos\; \backslash \; alpha$$

and $\backslash \; tan\; \backslash \; alpha\; \backslash \; sin\; \backslash \; theta\; (\backslash \; varphi)\; =\; \{D\; \backslash \; theta\; \backslash \; over\; D\; \backslash \; varphi\}$, nonlinear differential equation with separable variables in $\backslash \; theta\; (\backslash \; varphi)$

By separating the variables and while integrating between 0 and $\backslash \; varphi$:

$\backslash \; int\_\; \{\backslash \; pi/2\}\; ^\; \{\backslash \; theta\; (\backslash \; varphi)\}\{D\; \backslash \; theta\; \backslash \; over\; \backslash \; sin\; \backslash \; theta\}\; =\; \backslash \; tan\; \backslash \; alpha\; \backslash \; int\_0^\; \backslash \; varphi\; D\; \backslash \; varphi$

$\backslash \; ln\; \backslash \; left\; (\backslash \; tan\; \{\backslash \; theta\; (\backslash \; varphi)\; \backslash \; over\; 2\}\; \backslash \; right)\; =\; \backslash \; tan\; \backslash \; alpha\; \backslash \; varphi$ (cf Table of primitives)

$\backslash \; theta\; (\backslash \; varphi)\; =2\; \backslash ,\; \backslash \; operatorname\; \{Arctan\}\; \backslash \; left\; (\backslash \; mathrm\; \{E\}\; ^\; \{-\; \backslash \; varphi\; \backslash ,\; tan\; \backslash \; alpha\}\; \backslash \; right)$

The length L traversed is worth then, by definition:

$L=\; \backslash \; int\_0^\; \{+\; \backslash \; infty\}\; \backslash |f\text{'}\; (\backslash \; varphi)\; \backslash |D\; \backslash \; varphi$

where $f\text{'}\; (\backslash \; varphi)\; =\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; varphi\}\; (\backslash \; varphi,\; \backslash \; theta\; (\backslash \; varphi))\; +\; \backslash \; theta\text{'}\; (\backslash \; varphi)\; \backslash \; cdot\; \{\backslash \; partial\; \backslash \; vec\; M\; \backslash \; over\; \backslash \; partial\; \backslash \; theta\}\; (\backslash \; varphi,\; \backslash \; theta\; (\backslash \; varphi))$ and $\backslash |f\text{'}\; (\backslash \; varphi)\; \backslash |^2\; =\; \backslash \; sin^2\; \backslash \; theta\; +\; \backslash \; theta\text{'}^2\; =\; \backslash \; sin^2\; \backslash \; theta\; +\; \backslash \; tan^2\; \backslash \; alpha\; \backslash \; sin^2\; \backslash \; theta=\; \{\backslash \; sin^2\; \backslash \; theta\; \backslash \; over\; \backslash \; cos^2\; \backslash \; alpha\}$ and for the same reasons of sign, $\backslash |f\text{'}\; (\backslash \; varphi)\; \backslash |=\; \{\backslash \; sin\; \backslash \; theta\; \backslash \; over\; \backslash \; cos\; \backslash \; alpha\}$.

$L=\; \{1\; \backslash \; over\; \backslash \; cos\; \backslash \; alpha\}\; \backslash \; int\_0^\; \{+\; \backslash \; infty\}\; \backslash \; sin\; \backslash \; theta\; (\backslash \; varphi)\; D\; \backslash \; varphi$

While changing variable, with $\{D\; \backslash \; varphi\; \backslash \; over\; D\; \backslash \; theta\}\; =\; \{1\; \backslash \; over\; \backslash \; tan\; \backslash \; alpha\; \backslash ,\; \backslash \; sin\; \backslash \; theta\}$, one has $L=\; \{1\; \backslash \; over\; \backslash \; cos\; \backslash \; alpha\; \backslash \; tan\; \backslash \; alpha\}\; \backslash \; int\_0^\; \{\backslash \; pi\; \backslash \; over\; 2\}\; \backslash \; sin^2\; \backslash \; theta\; \backslash ,\; D\; \backslash \; theta$

$L=\; \{\backslash \; pi\; \backslash \; over\; 4\; \backslash ,\; \backslash \; sin\; \backslash \; alpha\}$

See too

• Orthodromy
• Navigation with the regard
• Road (navigation)

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