[Go] Map projection

1. Earth ellipsoid
The earth is a very complex surface of the sphere, people with the hypothetical average stationary sea surface formed by the "Earth body" as the reference, the approximation of the ellipsoid, the theory and practice proved that the ellipsoid is similar to a short axis of the Earth as the axis of the elliptical oval surface, the ellipsoid can be expressed in mathematical formula, The point on the natural surface is naturalized on this ellipsoid and can be calculated. Some of the commonly used ellipsoid and parameters are listed below:
1) Sea-Ford ellipsoid (1910) ellipsoid used before 52 in China
a=6378388m b=6356911.9461279mα=0.33670033670
2) Krasovsky ellipsoid (1940 Krassovsky) ellipsoid used in Beijing 54 coordinate system
a=6378245m b=6356863.018773mα=0.33523298692
3) 1975 I.U.G.G recommended ellipsoid (International Geodetic Association 1975) The ellipsoid used in XI ' an 80 coordinate system
a=6378140m b=6356755.2881575mα=0.0033528131778
4) WGS-84 ellipsoid (GPS Global Positioning System ellipsoid, 17 International Geodetic Association) WGS-84 coordinate system ellipsoid
a=6378137m b=6356752.3142451mα=0.00335281006247

The most commonly used geographic coordinate system is the latitude coordinate system, the coordinate system can determine the position of any point on the Earth, if we consider the Earth as an ellipsoid, and the graticule is added to the Earth's surface of the geographical coordinate frame of reference grid, longitude and latitude is from the Earth's center to the Earth's surface to the point of measurement of the angle, longitude is Latitude is the north-south direction, longitude crosses the Earth's north and South Poles, and parallels are the loops parallel to the equator. Geographical coordinates can be divided into astronomical geographical coordinates and geo-geographical coordinates: astronomical geographical coordinates are determined by astronomical surveying methods, geodetic coordinates are determined by geodetic method. The geographic coordinate system we use on the Earth's ellipsoid belongs to the Geo coordinate system, or the geodetic coordinate system.
After determining the size of the ellipsoid, it is also necessary to carry out ellipsoidal orientation, that is, to set the rotating ellipsoid in an appropriate position on the Earth, which is the "origin of coordinates" of the geographic coordinate system, and is the starting point of all geodetic coordinates calculation, commonly known as the "Earth origin".
It should be explained that the latitude coordinate system is not a planar coordinate system, because the degree is not a standard length unit, it is not used to measure the area length, planar coordinate system (also known as Cartesian coordinate system), because it has the following characteristics: can measure the horizontal x direction and vertical y-direction distance, can be measured in length, angle and area, Different mathematical formulae can be used to project the surface of the Earth sphere onto a two-dimensional plane. Each planar coordinate system has a specific map projection method.

2. Map projection
is to solve the contradiction that is depicted on the plane by the non-developable ellipsoid, by means of geometric perspective or mathematical analysis, the points and lines on the earth are projected onto the developable surface (planar, cylindrical or conical), and the developable surface is formed into a plane, and the corresponding relationship between the point and line on the plane and the point and line on the Earth's ellipsoid is established.
The process of map projection is to imagine a large enough piece of paper to wrap the Earth and project the Earth's objects onto this piece of paper. The surface of the earth is projected onto a plane, a conical surface or a cylindrical surface, and then the conical surface and the cylindrical face are cut along the bus to form a plane. According to the way this paper is wrapped, the map projection can also be divided into: azimuth projection, conic projection and cylindrical projection. According to the way this paper intersects with the Earth, the projection of the map can be divided into cutting projection and cutting projection, the figures on the tangent or secant are not deformed, and the distance tangent or secant deformation is larger.
A lot of projections are obtained directly by analytic method. Depending on the geometry of the plane can be divided into pseudo-azimuth projection, pseudo-conic projection, pseudo-cylindrical projection and so on.
There are two kinds of errors in the map projection, shape changes (also called angle changes) or area changes. It is important that the angle between the two figures on the map can be maintained in line with the field, since the projection, called the conformal projection (conformal mapping), which retains the shape of the projection after it has remained unchanged, is of great importance when navigating between the two places to keep the course accurate , or another example is the firing angle must be measured accurately, regardless of the long-range missile launch or short-distance projectile. So conformal projection is the most commonly used projection. The disadvantage of conformal projection is that the area of objects in high latitudes will be magnified. Projection can maintain the shape of the projection, known as the equal area projection (equivalent mapping), in the area of the need for analysis of the application is very important, the display of the relative area of the surface is accurate, but the shape will be changed, assuming that there is a circle on the Earth, the projection after the drawing becomes an ellipse. There is a third projection, non-conformal projection, meaning that both shape changes and area changes, such projections are unequal angle and unequal product, length, angle, area are deformed. Some of these projections hold a length ratio equal to 1 in one of the main directions, called equidistant projections.
Each projection has its own applicable aspects. For example, the Mercator projection is suitable for Huhaito, whose area deformation increases with the increase of latitude, but its direction is very small; the area deformation of the transverse Mercator projection increases with the distance from the central meridian, which is suitable for making different national maps. Conformal projections are commonly used for nautical charts, wind direction maps, ocean currents, and so on. This kind of projection is more used in topographic maps around the world now. The equal-product projection is used to plot economic areas and certain natural maps. For most mathematical maps and small-scale general maps, the requirement of equal product should be given priority. Geographical areas, such as the relative distribution of countries, waters and geographical classifications (vegetation, population, climate, etc.), are clearly very important elements. Arbitrary projections are often used as mathematical maps, and maps that require the right distance along a main direction. Commonly used as the world map projection has Mercator projection, the high-er projection, Moore's special projection, such as differential parallel multi-conic projection, the lattice projection, Sanson projection, Urmayev projection and so on. The following is a brief introduction to the Gaussian G-gauss–krüger projection used in topographic map of China.

2.1 Gauss-G Gauss–krüger Cartesian coordinates
The Gaussian-gram gauss–krüger projection (Gauss_krivger) belongs to the conformal transverse elliptic column projection, which is conceived to cross the outer part of the Earth ellipsoid with an elliptical column and tangent to the set central meridian. The longitude is perpendicular to each other, and the deformation is located at the intersection of the equator and the outer longitude of the projection belt, which is often used in higher latitude areas.
Gauss-gram Gauss–krüger projection belt stipulation: This projection is the mathematical foundation of the national Basic scale topographic map of our country, for controlling the deformation, adopting the method of the sub-band projection, in the scale 1:25,000-1:500,000 the chart uses the 6° band, the scale is 1:10,000 and greater than 1:10,000 of the figure uses the 3° band.
6° Band method: From Greenwich zero Longitude, each 6° is divided into a projection band, the world is divided into 60 projection belts, the eastern hemisphere from the east longitude 0°-6° for the first belt, the central meridian is 3°, and so on, the projection band number is 1-30. Its projection code n and the central meridian longitude L0 formula is: l0= (6n-3) °, the Western Hemisphere projection belt from 180° to 0 °, numbered 31-60, the projection code number n and the central meridian longitude L0 Calculation formula is l0=360-(6n-3) °.
3° Zoning method: From the longitude 1°30′, every 3 ° for the belt, the world divided into 120 projection belts, longitude 1°30′-4°30′, ... 178°30′-West longitude 178°30′, ... 1°30′-Longitude 1°30′.
The Eastern hemisphere has 60 projection bands, number 1-60, each with the central Meridian calculation formula: L0=3°n, the central meridian is 3°, 6° ... 180°
There are 60 projection belts in the Western Hemisphere, numbered 1-60, with the central Meridian calculation formula: L0=360°-3°n, the central meridian is 177 ° longitude 、... 3°, 0°.
China's provisions will be the vertical axis of each belt 500 km, will be all Y value plus 500 km, coordinate values before adding each band number. Taking 18 bands as an example, the original coordinate value is Y=243353.5m, the y=743353.5 after the westward shift, and the general coordinate of the band number is y=18743353.5.
In order to facilitate the retrieval of topographic maps of different scales, the paper introduces the distribution and numbering rules of topographic map in China.

Citation Link: http://www.cnblogs.com/wangzihao/archive/2010/09/22/1833048.html

[Go] Map projection