At present, the commonly used projections include Mercator projection (orthographic isometric cylindrical projection), Gauss-Kr Krüger projection (isometric transverse cylindrical projection), UTM projection (isometric transverse cylindrical projection), Lambert projection (isometric secant conical projection), etc.
Mercator projection
Mercator projection is a kind of "equiangular tangent cylindrical projection", which was drawn up by Dutch cartographer Mercator in 1569, assuming that the earth is enclosed in a hollow cylinder, its standard latitude line is tangent to the cylinder, and then assuming that there is a lamp in the center of the earth, projecting the figure on the sphere onto the cylinder, and then unfolding the cylinder. This is a map drawn by "Mercator projection" on a selected standard latitude line.
The Mercator projection has no angular deformation, the length ratio from each point to each direction is equal, its longitude and latitude lines are parallel straight lines, and intersect at right angles, the interval of longitude lines is equal, and the interval of latitude lines gradually increases from the standard latitude line to the two poles. On the map of Mercator projection, the length and area are obviously deformed, but the standard latitude line is not deformed, and the deformation gradually increases from the standard latitude line to the two poles, but because it has the characteristics of equal expansion in all directions, it keeps the correct direction and mutual position relationship.
The advantage of Mercator projection is to maintain the correct direction and angle on the map. Mercator projection maps are often used as navigation charts and aeronautical charts. If you follow the straight line between two points on the Mercator projection map, you can reach the destination without changing the direction. Therefore, it has favorable conditions for ships to locate and determine the course during navigation, which brings great convenience to navigators.
Gauss-Krüger projection
The Gauss-Kruger projection is an "equiangular crosscut cylindrical projection". Carl Friedrich Gauss (1777-1855), a German mathematician, physicist and astronomer, drew up the projection formula in the 1920s, which was later supplemented by Johannes Kruger (1857-1928), a German geodetic surveyor, in 1912, hence the name. It is assumed that a cylinder is used to cross the Center Meridian of the projection zone on the sphere. According to the conditions that the projection zone of the Center Meridian is a straight line with constant length and the equator projection is a straight line, the sphere within a certain range of longitude difference on both sides of the Center Meridian is projected to the cylinder. Then the cylindrical surface is sheared and flattened along the generatrix passing through the north and south poles to obtain the Gauss-Kruger projection plane.
After Gauss-Kruger projection, except Center Meridian and the equator are straight lines, all other meridians are curves symmetrical to Center Meridian. The Gauss-Kr Krüger projection has no angular deformation, and the deformation in length and area is also very small. The Center Meridian has no deformation, and the deformation increases gradually from the Center Meridian to the edge of the projection zone, and the maximum deformation is at the two ends of the equator in the projection zone. Because of its high projection accuracy, small deformation and simple calculation (the coordinates of each projection zone are the same, as long as the data of one zone is calculated, the other zones can be applied), its application in large-scale topographic maps can meet various military needs, and can accurately measure and calculate on maps.
Dividing the ellipsoid of the earth into several projection zones according to a certain longitude difference is the most effective method to limit the length deformation in the Gaussian projection. During zoning, it is necessary to control the length deformation so that it is not greater than mapping error, and to make the number of zones not too much so as to reduce the calculation work of changing zones. According to this principle, the ellipsoid of the earth is divided into melon-petal-shaped zones with equal longitude difference along the meridian, so as to facilitate zoning projection. It is usually divided into six degrees or three degrees according to the longitude difference of 6 degrees or 3 degrees. The six degree belt is divided from west to east every 6 degrees of longitude difference from the 0 degree meridian, and the belt numbers are numbered 1, 2.. 60 belts. The third degree belt is divided on the basis of the sixth degree belt. Its Prime Meridian coincides with the Prime Meridian of the sixth degree belt and the meridian of the belt, that is, the belt is divided from west to east every 3 degrees of longitude difference from the 1.5 degree meridian, and the belt numbers are numbered as the first, second and third degree belts in turn. 120 bands. The longitude range of China is from 73 ° in the west to 135 ° in the east, which can be divided into 11 zones of six degrees. The Center Meridian of each zone is 75 °, 81 °, 87 °.., 117 °, 123 °, 129 °, 135 °, or 22 zones of three degrees.
UTM projection
The full name of UTM projection is "Universal Transverse Mercator Projection", which is a kind of "Equiangular Transverse Secant Cylindrical Projection". The elliptical cylinder cuts the two contour circles of the earth at 80 degrees south latitude and 84 degrees north latitude. After projection, there is no deformation on the two secant meridians, while the length ratio on Center Meridian is 0.9996. UTM projection was created for global war needs, and the United States completed the calculation of this universal projection system in 1948. Similar to the Gauss-Kr Krüger projection, the projection angle is not distorted, the Center Meridian is a straight line, and is the axis of symmetry of the projection. The Scale Factor of Center Meridian is 0.9996 to ensure that there are two undistorted standard meridians about 330km away from Center Meridian. The UTM projection zoning method is similar to the Gauss-Kr Krüger projection, which divides the earth into 60 projection zones from west to east every 6 degrees of longitude difference from 180 degrees of west longitude. UTM projection is often used in Satellite Image data in China.
Lambert projection
Lambert's projection, also known as the "equiangular secant conic projection", was formulated by German mathematician Lambert (J. H. Lambert) in 1772. It is assumed that a right circular cone is tangent or secant to the spherical surface, and the spherical surface is projected onto the conical surface by applying the condition of equal angle, and then expanded along a generatrix, which is the Lambert projection plane. Aft projection, that latitude line is a concentric circular arc, and the longitude line is the radius of a concentric circle. Lambert projection uses two standard parallels to cut each other. Compared with the single standard parallel to cut each other, the projection deformation is small and uniform. The deformation distribution law of Lambert projection is as follows:
- The angle is not deformed, that is, the corresponding differential areas before and after projection remain similar, so it can also be called conformal projection.
- The equal deformation line is consistent with the latitude line, that is, the deformation on the same latitude line is equal everywhere.
- There is no distortion on the two standard parallels.
- On the same longitude, the outer side of the two standard latitudes is positive deformation (length ratio is greater than 1), while between the two standard latitudes is negative deformation (length ratio is less than 1). Therefore, the deformation is relatively uniform and the absolute value of the deformation is relatively small.
- The length of the line segment of the same latitude is equal, and the length of the longitude and latitude between two latitudes is equal everywhere.
Lambert projection is adopted in China's 1:1 million topographic map, and its division principle is consistent with the international one-millionth map projection stipulated by the International Geographical Society for global unified use. The latitude is divided into 15 projection zones from south to north according to the latitude difference of 4 degrees, the coordinates of each projection zone are calculated independently, each zone has two standard latitudes, the first standard latitude is the latitude at the south end of the map sheet plus the latitude of 30 ', and the second standard latitude is the latitude at the north end of the map sheet minus the latitude of 30', so that the coordinate results of each map sheet in the same projection zone are completely the same. The map deformation values of different zones are nearly equal, so it is only necessary to calculate the projection results of one map (latitude difference 4 °, longitude difference 6 °) in each projection zone. Due to the zonal projection with a latitude difference of 4 °, when the map is spliced along the latitude direction, no matter how many map sheets there are, there will be no cracks; however, when the map is spliced along the longitude direction, because the splicing line is located in different projection zones, the curvature after projection is different, resulting in cracks during splicing.
Related topics
Description of reference frame conversion method