main and collateral channels theory

The map coordinate system is determined by geodetic datum and map projection. Geodetic datum is the approximation of the earth's surface with a specific ellipsoid in a specific area, so every country or region has its own geodetic datum. The Beijing 54 coordinate system and Xi 'an 80 coordinate system we usually refer to actually refer to two geodetic datum lines in China. Referring to the former Soviet Union, China established Beijing 54 coordinate system and Krasovsky ellipsoid from 1953, and established China's new geodetic coordinate system-Xi 'an 80 coordinate system and IAG 75 earth ellipsoid recommended by the International Geodetic Association from 1978. At present, the results obtained by GPS positioning all belong to WGS84 coordinate system, WGS84 ellipsoid is used as the reference plane, and WGS84 is an ellipsoid. Therefore, relative to the same geographical location and different geodetic datum, their latitude and longitude coordinates are different.

The three ellipsoidal parameters adopted are as follows (extracted from Global Positioning System Measurement Specification GB/T18314-2001):

Ellipsoid long semi-axis and short semi-axis

Krassovsky 6378245 635638+088

IAG 75 6378 140 6356755.2882

WGS 84 6378 137 635638+042

Understanding: The ellipsoid is used to approach the earth, and it should be a vertical ellipse after rotation.

2. Geodetic datum

The relationship between ellipsoid and datum is one-to-many, that is, datum is based on ellipsoid, but ellipsoid cannot represent datum. The same ellipsoid can define different datum planes, such as Pulkovo 1942 in the former Soviet Union and Afgoye datum plane in Somalia, Africa, all of which adopt Krasovsky ellipsoid, but their datum planes are obviously different. In the current GIS commercial software, geodetic datum is defined by the transformation 7 parameters of WGS84-oriented local datum, that is, three translation parameters Δ x, Δ y and Δ z represent the translation values of two coordinate origins; The three rotation parameters εx, εy and εz respectively represent the rotation angles around Xt, Yt and Zt when the local coordinate system rotates to be parallel to the geocentric coordinate system. Finally, the scale correction factor is used to adjust the size of the ellipsoid. The conversion parameters of Beijing 54 and Xi 'an 80 relative to WGS84 have not been disclosed so far. In practical work, we can use the known Beijing 54 or Xi 'an 80 coordinate control points to convert the coordinate values of WGS84. When there is only one known control point (usually), the difference between Beijing 54 and WGS84 coordinates of the known point is taken as the translation parameter. When the working area is small, such as Qingdao, accuracy is enough.

Take (32, 12 1) as an example. The difference between the projection results of Beijing 54 and WGS84 in the north-south direction is about 63 meters (see the table below). For millions or millions of maps, this error is negligible, but it should be considered in engineering maps.

Input coordinates (degrees) Beijing 54 Gaussian projection (meters) WGS84 Gaussian projection (meters)

Latitude value (x) 32 3543664 354360 1

Longitude value (y)12121310994 21310997.

Understanding: the ellipsoid and the earth are definitely not completely fitted, so even if the same ellipsoid is used, different regions need to fit their own parts to the greatest extent because of different positions of concern, so the geodetic datum will be different.

3. Gaussian projection

Properties of (1) Gauss-Kruger Projection

Gauss-Kruger projection, which is called "Gauss Projection" for short, is also called "Isometric Transverse Elliptical Column Projection". It is a conformal projection between the ellipsoid and the earth plane. C.F.Gauss (1777- 1855), a German mathematician, physicist and astronomer, was drafted by Johannes Kruger (1857 ~ 1928) in the 1920s. According to the condition that the central meridian projection of the projection belt is a straight line with equal length and the equatorial projection is a straight line, the form of the function is determined and the Gaussian-Kruger projection formula is obtained. After projection, except the central meridian and equator are straight lines, all other meridians are curves symmetrical to the central meridian. Imagine an elliptic cylinder passing through the central meridian of the projection belt on an ellipsoid. According to the above projection conditions, the ellipsoid within a certain longitude difference range on both sides of the central meridian is orthographically projected onto the elliptic cylinder. The elliptic cylinder is cut and flattened along the generatrix passing through the north and south poles, which is the Gaussian projection plane. Taking the projection of the intersection of the central meridian and the equator as the origin, the projection of the central meridian as the ordinate X axis and the projection of the equator as the abscissa Y axis, a Gaussian Luger plane rectangular coordinate system is formed.

Gauss-Kruger projection has little deformation in length and area, and the central meridian has no deformation. From the central meridian to the edge of the projection belt, the deformation increases gradually, and the maximum deformation is at both ends of the equator in the projection belt. Because of its high projection accuracy, small deformation and simple calculation (the coordinates of each projection zone are consistent, so long as the data of one zone is calculated, the data of other zones can be applied), its application in large-scale topographic maps can meet various military needs, and it can be accurately measured and calculated on the map.

(2) Gauss-Kruger projection zoning

Dividing the earth ellipsoid into several projection zones according to certain differences is the most effective method to limit the length deformation in Gaussian projection. When zoning, the length deformation should be controlled to be less than the mapping error, and the number of zoning should not be too much to reduce the calculation of changing zoning. According to this principle, the earth ellipsoid is divided into melon-petal-shaped partitions with equal meridian difference, and the partition projection is carried out. Usually, it is divided into six-degree bands or three-degree bands according to the difference of six degrees or three degrees. The 6-degree zone is divided into several zones from west to east every 6 degrees from the 0-degree meridian, and the area codes are 1 and 2…60 in turn. The third-degree zone is divided on the basis of the sixth-degree zone, and its central meridian coincides with the central meridian and zoning meridian of the sixth-degree zone, that is, it is zoned from west to east every three degrees from the meridian of 1.5, and the area codes are sequentially numbered as 1 and 2… 120 zones of the third-degree zone. The longitude of China varies from 73 in the west to135 in the east, which can be divided into eleven zones of six degrees. The central meridian of each region is 75, 8 1, 87, ..., 1 17, 123, 65438 in turn. Six-degree zone can be used for small and medium-scale mapping (such as 1: 250000), and three-degree zone can be used for large-scale mapping (such as 1: 10000). Three-dimensional Gaussian projection is often used in urban construction coordinates.

(3) Gauss-Kruger projection coordinates

Gauss-Kruger projection is projected separately according to the partition method, so the coordinates of each partition form an independent system. Taking the central meridian projection as the vertical axis (X), the equatorial projection as the horizontal axis (Y), and the intersection of the two axes as the coordinate origin of each belt. The ordinate is calculated from the zero equator, positive to the north and negative to the south. China is located in the northern hemisphere, and its ordinate is positive. If the abscissa starts from the central meridian, the east of the central meridian is positive, the west is negative, and the abscissa is negative, it is inconvenient to use. Therefore, it is stipulated that the coordinate longitudinal axis moves 500 kilometers to the west as the starting axis, and all the abscissa values in the belt add up to 500 kilometers. Because the coordinates of each projection zone in Gaussian-Kruger projection are relative to the origin of local zone coordinates, the coordinates of each zone are exactly the same. In order to distinguish which area a coordinate system belongs to, add a band number before the horizontal coordinate, such as (423 1898m, 2 1655933m), where 2 1 is the band number.

(4) Gauss-Kruger projection and UTM projection

Some foreign software, such as ARC/INFO or the supporting software of foreign instruments, such as multi-beam data processing software, often support UTM projection instead of Gaussian-Kruger projection, so UTM projection coordinates are often submitted as Gaussian-Kruger projection coordinates.

UTM projection is called "Universal Transverse Mercator Projection" and is an equiangular transverse tangent cylindrical projection (Gauss-Kruger is an equiangular transverse tangent cylindrical projection). The cylinder is tangent to the earth in two equal-height circles of 80 degrees south latitude and 84 degrees north latitude. This projection divides the earth into 60 projection zones, each with a longitude difference of 6 degrees, which has been used as the mathematical basis of topographic maps in many countries. The main difference between UTM projection and Gaussian projection lies in the scale coefficient of north-south grid lines. The central meridian length of Gauss-Kruger projection remains unchanged after projection, that is, the scale coefficient is 1, while the scale coefficient of UTM projection is 0.9996. The scale coefficient of UTM projection is constant along each north-south grid line, but variable in the east-west direction. The scale coefficient of the central grid line is 0.9996, which is about 363 kilometers away from the center point of the widest edge of the north-south vertical line, and the scale coefficient is 1.00 158.

Gauss-Kruger projection and UTM projection can be approximately transformed by Xutm=0.9996 * X Gaussian and Yutm=0.9996 * Y Gaussian. The following is an example (the reference plane is WGS84):

Input coordinates (degrees) Gaussian projection (meters) UTM projection (meters) Xutm=0.9996 * X Gaussian, Yutm=0.9996 * Y Gaussian.

Latitude value (x) 32 3543600.9 3542183.5 3543600.9 * 0.9996 ≈ 3542183.5.

Longitude value (y)12121310996.831072.4 (310996.8-500000)

Note: the coordinate point (32, 12 1) is located in the 2 1 band of Gaussian projection, and the first two digits of the Gaussian projection y value 2 13 1096.8 are signed; The coordinate point (32, 12 1) is located in the 5 1 band of UTM projection, and the Y value of UTM projection in the above table is not marked. Because the coordinate axis has moved 500,000 meters to the west, you must subtract 500,000 from the y value, multiply it by the scale factor, and then add 500,000.

Understanding: The method of Gaussian projection is to keep the equator and central meridian unchanged and flatten the sphere. Methods: Cover the ellipsoid with an elliptical column, project it onto the elliptical column, and then open the elliptical column.

Note: the coordinate point (32, 12 1) is located in the 2 1 band of Gaussian projection, and the first two digits "2 1" in the Gaussian projection y value 2 13 1096.8 are signed; The coordinate point (32, 12 1) is located in the 5 1 band of UTM projection, and the Y value of UTM projection in the above table is not marked. Because the coordinate axis has moved 500,000 meters to the west, you must subtract 500,000 from the y value, multiply it by the scale factor, and then add 500,000.

Understanding: The method of Gaussian projection is to keep the equator and central meridian unchanged and flatten the sphere. Methods: Cover the ellipsoid with an elliptical column, project it onto the elliptical column, and then open the elliptical column.

(4) Gauss-Kruger projection and UTM projection

Some foreign software, such as ARC/INFO or the supporting software of foreign instruments, such as multi-beam data processing software, often support UTM projection instead of Gaussian-Kruger projection, so UTM projection coordinates are often submitted as Gaussian-Kruger projection coordinates.

UTM projection is called "Universal Transverse Mercator Projection" and is an equiangular transverse tangent cylindrical projection (Gauss-Kruger is an equiangular transverse tangent cylindrical projection). The cylinder is tangent to the earth in two equal-height circles of 80 degrees south latitude and 84 degrees north latitude. This projection divides the earth into 60 projection zones, each with a longitude difference of 6 degrees, which has been used as the mathematical basis of topographic maps in many countries. The main difference between UTM projection and Gaussian projection lies in the scale coefficient of north-south grid lines. The central meridian length of Gauss-Kruger projection remains unchanged after projection, that is, the scale coefficient is 1, while the scale coefficient of UTM projection is 0.9996. The scale coefficient of UTM projection is constant along each north-south grid line, but variable in the east-west direction. The scale coefficient of the central grid line is 0.9996, which is about 363 kilometers away from the center point of the widest edge of the north-south vertical line, and the scale coefficient is 1.00 158.

Gauss-Kruger projection and UTM projection can be approximately transformed by Xutm=0.9996 * X Gaussian and Yutm=0.9996 * Y Gaussian. The following is an example (the reference plane is WGS84):

Input coordinates (degrees) Gaussian projection (meters) UTM projection (meters) Xutm=0.9996 * X Gaussian, Yutm=0.9996 * Y Gaussian.

Latitude value (x) 32 3543600.9 3542183.5 3543600.9 * 0.9996 ≈ 3542183.5.

Longitude value (y)12121310996.831072.4 (310996.8-500000)

Note: the coordinate point (32, 12 1) is located in the 2 1 band of Gaussian projection, and the first two digits of the Gaussian projection y value 2 13 1096.8 are signed; The coordinate point (32, 12 1) is located in the 5 1 band of UTM projection, and the Y value of UTM projection in the above table is not marked. Because the coordinate axis has moved 500,000 meters to the west, you must subtract 500,000 from the y value, multiply it by the scale factor, and then add 500,000.

Understanding: The method of Gaussian projection is to keep the equator and central meridian unchanged and flatten the sphere. Methods: Cover the ellipsoid with an elliptical column, project it onto the elliptical column, and then open the elliptical column.

Note: the coordinate point (32, 12 1) is located in the 2 1 band of Gaussian projection, and the first two digits "2 1" in the Gaussian projection y value 2 13 1096.8 are signed; The coordinate point (32, 12 1) is located in the 5 1 band of UTM projection, and the Y value of UTM projection in the above table is not marked. Because the coordinate axis has moved 500,000 meters to the west, you must subtract 500,000 from the y value, multiply it by the scale factor, and then add 500,000.

Understanding: The method of Gaussian projection is to keep the equator and central meridian unchanged and flatten the sphere. Methods: Cover the ellipsoid with an elliptical column, project it onto the elliptical column, and then open the elliptical column.