Rhumb line

The rhumb line ( gr Loxos " wrong," dromos " running ") is a curve on a spherical surface, such as the earth's surface that intersects the meridians in the geographic coordinate system always at the same angle and is therefore also called rhumb same angle or constant curve course.

More generally, for every rotating body a rhumb line as constant curve course that the ball is, especially Kugelloxodrome, the rhumb line of the cylinder, the helix, the cone of the conical spiral (or conical helix).

They were discovered in 1550 by Pedro Nunes, the name comes from Willibrord Snell ( 1624).

Properties

Except in special cases, cutting angles 0 ° and 90 ° cutting angles, the rhumb line is not closed. It spirals around the earth and thereby approaches the poles at. In the strictly mathematical sense, the rhumb line while never reached the Pole, but approaches him only asymptotically, while infinitely often winds around the polar region.

In the special case of a cut angle with the meridian of 0 ° is the rhumb line itself is a meridian and thus great circle, that passes through the poles. This is also the only special case of a rhumb line that reaches the pole. It follows by implication: Since solely the rhumb line 0 ° reached the North Pole, conversely, starts only from the North Pole alone and the rhumb line 180 °. Thus, the geographic North Pole to get only in the direction 180 ° away - but the course information is 180 ° at the North Pole is not defined, it could be with this course move from the North Pole on each Meridian - to Moscow or Los Angeles or ... Only the arrival at the South Pole is guaranteed. In practical navigation, this problem is circumvented by in high latitudes after the grid navigation ( engl. grid navigation ) is navigated with polar stereographic cards.

In the second special case - cutting angle 90 ° - the rhumb line is also closed, forms a parallel of latitude, and is therefore generally not a great circle. The only latitude that is a great circle is the special case of the equator, so when on the rhumb line to latitude constant is 0 °.

Projections:

• In cartography the rhumb line are shown as straight lines on maps in the Mercator projection.
• In a stereographic projection of the curve is a logarithmic spiral
• In an orthographic azimuthal (parallel cracks along the Earth's axis ) creates a spiral Poinsotsche

In polar regions the rhumb line the properties of a (plane ) has thus locally spiral, near the equator, but properties of a helix ( spiral spatial ).

Rhumb line to the pole

Rhumb line at the equator

Calculation

The formula of the rhumb line ( the slope angle in the projection ) is derived from the above-mentioned property of the Mercator projection, map rhumb lines as straight lines.

• The length with
• Denotes the width with

To the west towards the east is negative, positive; is positive for latitudes of the northern hemisphere and negative in the southern hemisphere. Both angles are used here mathematically in radians, not in degrees

• The direction angle is a constant bearing in the direction of true north, the slope angle in the Mercator projection, with the slope

In spatial coordinates

Be the longitude coordinate of any point of the rhumb line ( which is not limited to ).

In the Mercator projection, the curve is a straight line:

For a point of the width and length due to the mapping rule applies Mercator projection

For the width of the point thus results or, expressed as Gudermann function gd, In Cartesian coordinates, with a radius of a sphere:

This curve is in longitude by the equator, for any positions of the equator passage and in the above formulas, the term is to be replaced accordingly.

In the Mercator projection

In the air and in particular the seafaring it may be convenient to travel along a rhumb line, because then you always have to follow only one bearing (compass direction). Although the route of the rhumb line is always longer than the geodesic (only if the rhumb line lies on a great circle, they can be the same length ) - for that you must not constantly calculate a new heading angle. At shorter distances, the navigation on the rhumb line is only slightly longer than the navigation on the geodesic. In air transport the Lambert Conformal Conic projections are used.

Mercator projection forming a point with the coordinates on the coordinate plane on which:

Mercator projection the two points and a right-angled triangle with a hypotenuse of the right angle and is formed in the projection plane. For the angle at follows:

Using the two -digit function the Cartesian to the coordinates and the angle of the polar coordinate representation provides and stands as atan2 function in most programming languages ​​, we obtain:

The direction angle of the rhumb line, which is calculated from north through east clockwise, then:

The route, which is - within the Mercator map - travels between points A and B on the rhumb line is:

It should be noted that this is only the shortest rhumb line if and only if, they so westwards farther apart than in an easterly direction, in the other case this is only the second best way. In addition, can be between any two points (except the poles) always find any number of rhumb lines, which then circle one or more times the ball ( earth). For these cases in the initial equation for a value other side of the tangent to choose. In the Mercator projection of the graph migrates beyond the right or left edge and appears on the other side.