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Rhumb Line and Great Circle
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Rhumb Line and Great Circle

Rhumb Line and Great Circle.

 

rhumb lineOr travel curve equal angles is a line that intersects the meridians at the same angle.

On the surface of the globe rhumb line has the form of the spatial spiral, with each revolution around the globe asymptotically approaching the pole.

In addition to the special cases when rhumb line and orthodromy match (flight along the meridian or equator), longer than the great circle and rhumb line is convex always to the equator.

Maximizing the way when flying rhumb line in comparison with the great circle flight, depending on the difference in longitude IPM n CPM (H2 Z ^), and the latitude at which the increase in the maximum way.

Flights by rhumb now, especially on airplanes with GTE, practical applications have not, and so the rhumb line laying on the flight map here does not understand. It should only be noted that Rhumb great length must be routed through the points. To do this, you must first calculate the track angle and the coordinates of its intermediate points. Elements rhumb line can be calculated according to the formulas or define graphically using the graticule in the Mercator projection.

Asking longitude waypoint, find the value & D, are a function of latitude. Latitude f intermediate point D is found by using special tables annexed to textbooks aeronautical cartography.

orthodrome

orthodrome It called line of shortest distance between two points on the earth's surface. Great Circle is an arc of a great circle whose plane passes through the center of the world and the two given points on the surface of the globe. Meridians are great circle connecting the north and south geographic pole, - are special cases of the Great Circle. Generally orthodromy intersects at different meridians, unequal angles between them.

The reference meridian (OM) - the meridian passing through the starting point of the Great Circle area. Calculations or measurements orthodromic direction of the airplane is performed from the reference meridian.

 

Rhumb Line and Great Circle 2

 

Equation rhumb line to the surface of the globe

 

In addition to the special cases when rhumb line and orthodromy match (flight along the meridian or equator), longer than the great circle and rhumb line is convex always to the equator.

Maximizing the way when flying rhumb line in comparison with the great circle flight, depending on the difference in longitude IPM n CPM (H2 Z ^), and the latitude at which the increase in the maximum way.

Flights by rhumb now, especially on airplanes with GTE, practical applications have not, and so the rhumb line laying on the flight map here does not understand. It should only be noted that Rhumb great length must be routed through the points. To do this, you must first calculate the track angle and the coordinates of its intermediate points. Elements rhumb line can be calculated according to the formulas or define graphically using the graticule in the Mercator projection.

Asking longitude waypoint, find the value & D, are a function of latitude. Latitude f intermediate point D is found by using special tables annexed to textbooks aeronautical cartography.

Great Circle is a line of the shortest distance between two points on the earth's surface. Great Circle is an arc of a great circle whose plane passes through the center of the world and the two given points on the surface of the globe. Meridians are great circle connecting the north and south geographic pole, - are special cases of the Great Circle. Generally orthodromy intersects at different meridians, unequal angles between them.

The reference meridian (OM) - The meridian passing through the starting point of the Great Circle area. Calculations or measurements orthodromic direction of the airplane is performed from the reference meridian.

Ortodromicheskny track angle -the angle formed by the northern direction of the reference meridian and the line of the path.

The initial azimuth of the great circle (A) - the angle formed by the northern direction of the meridian passing through the starting point of the Great Circle and the Great Circle.

When flying on a great circle should be remembered that the IMPs great circle is changed to angle B - the angle of convergence of the meridians, which can be calculated from the approximate formula

Elements of Great Circle (length, location of the intermediate points and ortodromicheskny track angle) can be calculated according to the formulas of spherical trigonometry or graphically.

3-y way. GTC can be measured directly on the map from any meridian followed by an amendment to the convergence angle of the reference meridian and the meridian angle measurement space.

To perform the desired course for the flight in the opposite direction stopped orthodromic travel angle is derived from the meridians, the former end when flying in the original direction. Consequently, the travel angle during the flight there and back are different from each other not only on 180 °, but also by the amount of correction for the convergence of the meridians.

 

If the entire flight route runs along the Great Circle and has PG1M, the calculation of the travel angle is greatly simplified:

  • 1) route is divided into sections according 1 000-1200 km.

  • 2) meridians starting points plots find support for both directions and from the measured angles orthodromic travel.

 

Image Great Circle on maps in different projections. The maps drawn up conformal cylindrical projection (Mercator), orthodromy depicted as a complex curve, always facing bulge in the geographic poles.

The maps of central projection all the great circle represented by a straight line.

The maps in the polar stereographic projection orthodromy generally depicted circular arc. The curvature of the great circle is smaller, the closer it is to the geographic pole. Meridians, which are a special case of the Great Circle, polar stereographic projection maps are represented by straight lines.

On-board aeronautical 1 scale map: 2 000 000, prepared in a modified Polyconic, orthodromy within one sheet is almost a straight line.

Line of equal azimuth (LRA), or equal to the radio bearing is called the line at each point where a ground station bearing the same angle (YPR). With the lines of equal azimuth lines as the provisions define the location of the aircraft on the ground pulse beacons using the vehicle's radar line of equal azimuth - a complex curve. On the globe it crosses the meridian at different angles, and only with the meridian passing through the installation point of the station, it is an angle equal to the true radio bearing (IPR).

(LRR) is a line on the earth's surface, all points of which are from a certain point at the same distance - on the circumference of a small circle of the globe. As the state line, the line of equal distance in piloting used for astronomical measurements of the height of the luminaries using the sextant. In aviation astronomy LRA is called the circle of equal altitude, on a map, it is replaced by a tangent - direct equal heights. The elements of this line is calculated with the help of special tables of altitudes and azimuths of celestial bodies - TBA.

LRR used in the application of angle-ranging and bipolar radio ranging systems.

On the maps of different projections LRR have a different view. The maps LRR stereographic projection - circumference.

LRR is applied to a card with a high degree of accuracy for intermediate LRR points calculated according to the formulas of direct and inverse geodetic problems (the calculations are made on the surface of the ellipsoid Krasovsky).

Hyperbole (spherical), or a line of equal distance difference is the curve at each point where the difference between the distances to two fixed points (station) is a constant. For the purposes of piloting using lines of equal distance differences exist hyperbolic navigation system. Such systems include two pairs (or chain) of ground stations and airborne equipment (transceiver indicators), can adequately measure the difference between the distances from the aircraft to the radio stations.

Place the plane on a hyperbolic system is determined by the intersection of two hyperbolas.

Map - Conditional reduced generalized built according to certain mathematical rules image the Earth's surface on a plane.

Plan - Image plane in small scale earth surface sections taken through the plane.

map projection - A way to image the Earth's surface or the earth's ellipsoid on a plane.

Scale - The ratio of the length of the line on the map to the surface length of the corresponding line of the Earth.

The main scale (M) It shows how many times reduced the globe (or an ellipsoid) in the design of it on the plane. The main scope is always indicated on the map.

Private scale (c) defined as the ratio of an infinitesimal segment on the map at a given point in this direction and the corresponding infinitesimal segment on the surface of the globe (or ellipsoid).

particular scale in the direction of the meridian indicated by the letter T, and in the direction of the parallels - the letter n.

The main directions are called the directions in which the scale of private or minimal, maximal IPT. The maximum and minimum scale at this point indicated by a and b. Almost all projections cards used in aircraft navigation, the principal directions coincide with the meridians and parallels.

 

Zoom in by the ratio of private to the main scale

 

At length distortion determined by the difference between the increase in the scale and unit.

Distortion with directions given by the difference between the direction of the globe and in the same direction on the map. Maximum distortion of the areas at a given point is calculated by the formula

P is called the scale of the area ratio of the infinitely small area on the map to the corresponding area on the surface of the globe, which is reduced to the size of the globe before designing it on the plane. The distortion of space is characterized by the scale of the area.

By the nature of the distortion katograficheskie projections are divided into conformal, Equidistant, of equal and arbitrary.

Conformal projections are characterized by the fact that the angles and directions on maps drawn up in these projections are shown without distortion; Real scale of the main directions are equal; infinitesimal figure retain the likeness on the map corresponding figures in the world. These data are expressed as follows: a - b; s = 0; p = ab.

Conformal projection allows more easily determine the direction and therefore are widely used to create aeronautical charts, forpiloting important precise measurement of direction.

Called equidistant projection, in which private scale all points on one of the main areas are the main scale.

Equal-area projection is called, in which the area of ​​the depicted figure equal to the area of ​​the same shape on the map.

Called arbitrary projections, which are not equal angles are not equal, intermediate and equal area.

Custom projections are virtually very little distortion in the directions of length and area, and therefore are widely used in aircraft navigation. Aeronautical Chart scale 1: 2 000 000 prepared in a modified Polyconic which is arbitrary.

Depending on the type of normal grid or a method of constructing graticule projection maps used in aircraft navigation divided into cylindrical, conical, sex and conical, azimuthal et al. Normal called this grid coordinate lines corresponding to a certain coordinate system, which has the simple image Dinah projection. Some projections normal grid coincides with the geographic grid.

Cylindrical projection - a projection in which the meridians of normal grid are represented by straight lines parallel to each other and spaced at a distance proportional to the difference between the corresponding longitudes; parallels are depicted as

straight lines perpendicular to the meridians. This grid is obtained by designing a grid of meridians and parallels of the globe to the side surface of the cylinder (tangent or secant) and the deployment of the surface plane of the PA.

Depending on the location of the cylinder axis relative to the axis of rotation of the globe cylindrical projection is divided into a normal cylindrical projection (cylinder axis coincides with the axis of rotation of the globe), the transverse cylindrical projection (cylinder axis perpendicular to the axis of rotation of the globe), oblique cylindrical projection (the angle between the axis of rotation of the cylinder and the axis Globe longer and less 0 90 °).

In normal cylindrical projections normal grid coincides with the geographic grid of meridians and parallels.

Simple cylindrical projection has the following equation of rectangular coordinates.

View geographic grid in a simple cylindrical projection: Meridian - straight lines parallel to each other and spaced at a distance proportional to the difference between the N longitude; Parallel - straight, perpendicular to the meridians, spaced at a distance proportional to the difference in latitude.

The projection of equidistant lines of the meridians. All parallels (except the equator) are distorted. Distortions in the direction parallel increase with increasing latitude. On the pole is the maximum distortion, since the point of the poles are represented by straight lines, the length of which is equal to the length of the equator. Distortion angles and areas are also increasing with increasing latitude. In extreme angular distortion (2so) reaches 180 °, and the scale of the square is equal to infinity.

Near the equator (in the band p <± 5 °) projection of nearly conformal, equal area and equal-between.

Conformal cylindrical projection of Mercator. Equations Cartesian coordinates of points in conformal cylindrical n roektsii.

View geographical grid: the meridians are shown as well as in a simple cylindrical projection; Parallel - straight, perpendicular to the meridians; the distance between the parallels with increasing latitude increases in proportion to the difference between the meridional parts.

Conformal projection. The distortion of lengths proportional to the secant of the latitude. The distortion of space is proportional to the square of the secant latitude.

The maps conformal cylindrical Mercator rhumb line is always represented by a straight line across the meridians at a constant angle. Near the equator in the band p <± 5 ° projection of nearly conformal, equidistant and equal area.

Conformal cross - cylindrical projection Gauss obtained as a result of the design of the ellipsoid to cylinder touches of a meridian (the axis of rotation of the ellipsoid and the cylinder intersect at an angle 90 °).

This projection band ellipsoid bounded meridians multiples 6 °, projected on a cylinder in its system flat.

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