1. Background

In understanding these two conversion methods, it is necessary to first understand some of the basic knowledge related to this. We have three common ways to represent spatial coordinates: latitude and high, planar and high, and spatial cartesian.

2. Latitude and longitude coordinate system (geodetic coordinate system)

The first thing I want to emphasize here is that the latitude and longitude of astronomical coordinates is different from that of the geodetic coordinate system. So, the same latitude and longitude values, in the BJ54 and WGS84 are different positions, and the following I said the latitude and longitude are the geodetic coordinates of the latitude and longitude. Geodetic coordinate system is a coordinate system established by reference ellipsoid as Datum plane in geodetic survey. Let me give you a general discussion of the two important concepts involved.

2.1 GeoID and Earth Sphere

The surface of the Earth itself is an uneven, irregular surface that cannot be expressed in mathematical formulas and cannot be operated on, so we must find a regular surface in place of the natural surface of the earth in measurement and mapping.

When the ocean is stationary, its free water surface must be orthogonal to the gravity direction of the points on the plane (lead vertical direction), which we call the geoid. However, there are countless geoid on the Earth, and we envision the geoid, which coincides with the stationary average sea surface, as a closed-form that can wrap the earth, which is the geoid.

And the sphere formed by the geoid parcel is the **Earth sphere** .

2.2 Earth Ellipsoid

Because of the uneven mass distribution in the Earth body, which causes the change of the direction of gravity, the geoid edge which is orthogonal to the gravity direction becomes a very irregular surface which can not be expressed mathematically. But although the geoid is very irregular in shape, it is already a very close to the spheroid that rotates around the axis of rotation (the short axis).

Therefore, in the measurement and mapping using the spheroid to replace the Earth sphere, the rotating sphere is often called the **earth ellipsoid** , referred to as **ellipsoid** .

2.3 Common geodetic coordinate systems

In different coordinate systems, the long radius of the ellipsoid, the short radius and the flattening rate are different. For example, the ellipsoid that corresponds to the four coordinate systems we use, their ellipsoid parameters are different:

BJ54 coordinate system: belongs to the heart coordinate system, the long axis 6378245m, the short axis 6356863, the flat rate 1/298.3.

XIAN80 coordinate system: belongs to the heart coordinate system, the long axis 6378140m, the short axis 6356755, the flat rate 1/298.25722101.

WGS84 coordinate system: belong to geocentric coordinate system, long axis 6378137.000m, short axis 6356752.314, flat rate 1/298.257223563.

CGCS2000 coordinate system: The geocentric coordinate system, the long axis 6378137.000m, the short axis 6356752.31414, the flat rate F = 1/298.257222101.

From the above parameters we can see that for the geodetic coordinate system, we can also be divided into two types: the core coordinate system and the geocentric coordinate system.

2.3.1 Centroid coordinate system

"Shen Xin" means the center of the reference ellipsoid. In the measurement, in order to deal with the observation results and to calculate the coordinates of the ground control network, it is usually necessary to select a reference ellipsoid as the basic reference plane, select a reference point as the starting point of geodetic survey (Earth origin), and use astronomical observations of the Earth origin to determine the position and direction of the reference ellipsoid within the Earth. The application of the geodetic coordinates is very extensive, it is a universal coordinate system of the classical Geodetic Survey. According to the theory of map projection, the reference geodetic coordinate system can be converted into a plane Cartesian coordinate system by Gauss projection calculation, which provides a control basis for topographic surveying and engineering surveying.

The BJ54 coordinate system is a coordinate system which is based on the Krasovsky ellipsoid and is produced by local adjustment. Its Earth origin in the former Soviet Union Pulkovo, longitude 30°19′15 ", north latitude 59°46′6".

The XIAN80 coordinate system is the Earth ellipsoid as recommended by the 1975 International Geodetic and Geophysical Federation at its 16th session. Its Earth origin in central China's Shaanxi province Jingyang County Yongle Town, east longitude 108°55′25.00″, north latitude 34°32′27.00″.

2.3.2 Geocentric coordinate system

The geocentric coordinate system (geocentric coordinate system) is a spatial Cartesian coordinate system established at the origin of the Earth's centroid, or a geodetic coordinate system based on the Earth's ellipsoidal surface coincident with the centroid of the Earth's center.

The reason for this is that the early-riser coordinate system, although it can well meet the local measurement needs, but can not meet the needs of global positioning, because the core coordinate system with the local geoid is the most consistent ellipsoid as a reference ellipsoid, and this reference ellipsoid does not meet the needs of global positioning. So in the geocentric coordinate system, we use the ellipsoid which is the most dense with the global geoid as the reference ellipsoid.

At present, the measurement of geocentric coordinate system is mainly three kinds of methods: gravity measurement, satellite geodetic method and local coordinate conversion. Both GPS positioning and Beidou positioning are known as satellite geodetic measurements.

2.3.3 Inference

A. geocentric coordinate system is more suitable for global measurement.

B. The coordinates under the WGS84 are basically the same coordinates under the CGCS2000 coordinate system, and the precision gap is centimeter-level.

C. The transformations in the same ellipsoid are strict, and the conversion between the different ellipsoid is not strict. For example, between the WGS-84 coordinates and the Beijing 54 coordinate there is no set of conversion parameters that can be used nationally, and will be different in each place, as they are two distinct ellipsoid benchmarks.

3. Planar coordinate system (projected coordinate system)

First we must make it clear that the projected coordinate system is built on top of the geographic coordinate system. In other words, you must have coordinates in a geographic coordinate system before you can project the coordinates to get the projected coordinates.

So why is there a geographic coordinate system and a projection coordinate system?

Because the degree of latitude and longitude does not correspond to a standard length, it is not possible to accurately measure distances or areas, and it is difficult to display data on a flat map or computer screen. When using many (but not all) GIS analysis and mapping applications, it is often necessary to have a more stable planar coordinate framework provided by the projected coordinate system. Unlike geographic coordinate systems, the length, angle, and area of a projected coordinate system are constant within a two-dimensional space. The projected coordinate system is always based on a geographic coordinate system, while the latter is based on a sphere or spheroid. In the projected coordinate system, the location is identified by the X, Y coordinates of the grid, and its origin is in the grid center.

According to the method of projection, we divide the projection into two categories: geometric projection and non-geometrical projection. In our country, the most common projection is Gaussian g gauss–krüger projection, the basic Chinese sub-provincial (area) map projection and large-scale map projection are selected as Gaussian gram Gauss–krüger projection. Moreover, various large and medium scale topographic maps use different Gaussian-gram gauss–krüger projection bands. The topographic map of which greater than 1:10,000 uses 3° belt, 1:25,000 to 1:500,000 of the topographic map adopts the 6° belt.

3.1 Gauss G Gauss–krüger projection (Gauss_kruger)

It is assumed that an elliptical cylinder is cross-nested outside the Earth ellipsoid and tangent to a certain meridian (this meridian is called the Central Meridian or Meridian), and the center axis of the elliptic column is centered through the ellipsoid, and then a certain projection method is used to project the regions of the central meridian on each side of the range to the elliptic cylinder surface. Expanding this cylinder becomes the projection plane, and the projection is a Gaussian projection.

Gaussian projection is a kind of normal projection, which belongs to the geometric projection.

On the projection surface, the projections of the central meridian and the equator are straight lines, with the central meridian and the equator

The intersection of 0 as the origin of the coordinates, the projection of the central meridian is the ordinate x axis, and the projection of the equator is the Y axis. In China, the x-coordinate is positive, and the maximum y-coordinate (6° band at the equator) is about 330km. To avoid negative horizontal axis, add 000m to the horizontal axis. You should also prefix the coordinates with a band number. This coordinate is called the National uniform coordinate. For example, there is a bit of y=19 623 456.789m, which is located in the 19 band, east of the central Meridian, and its horizontal axis relative to the central meridian is: First remove the band number, minus 000m, and finally the =123 456.789m.

And, due to the use of the band method, the projection of each band is exactly the same, a coordinate value (x, y), in one of the projections, there are 60 of the same coordinate values in the world, can not exactly indicate the location of the point. Therefore, before the Y-value, it needs to be labeled with a band number, and such coordinates are called universal coordinates.

Comparison of 3.2 Gauss G Gauss–krüger projection with UTM projection

The UTM projection, called Universal Transverse mercatol PROJECTION (Universal Transverse Mercator), is a "conformal transverse cylindrical projection", which cuts the Earth at 80 degrees south latitude, 84 degrees north latitude, two, and other high circles, After projection, the two tangent meridians are not deformed, and the length of the central meridian is 0.9996. The UTM projection was created for global warfare, and the United States completed the calculation of this universal projection system in 1948. Similar to the Gaussian-gram gauss–krüger projection, the projection angle is not deformed, the central meridian is a straight line, and is a projection of the symmetrical axis, the central meridian of the scale factor of 0.9996 is to ensure that the central meridian around 330km there are two non-distorted standard meridians.

The UTM projection band method is similar to the Gaussian-gram gauss–krüger projection, which divides latitude 84 degrees to 80 degrees latitude by longitude into 60 bands, each with 6 degrees. From 180 degrees west longitude, two standard parallels are about 180Km from the central meridian, and the central meridian has a ratio of 0.9996. UTM projection is often used in satellite imagery data in China.

From the perspective of projection geometry, the Gaussian-gram gauss–krüger projection is "conformal transverse cylindrical projection" (Transverse tonformal tylinder Projection), the central meridian remains unchanged after projection, that is, the scale factor is 1;UTM projection is "conformal transverse cylindrical projection", Cylindrical cut the Earth at 80 degrees south latitude, 84 degrees north latitudes two, such as High circle, the projection after the two secant no deformation, the central meridian length than 0.9996. Therefore, if the same ellipsoid is used, the main difference between the two is the proportional factor, the ratio coefficient of Gaussian-G Gauss–krüger projection central meridian is 1, the UTM projection is 0.9996, the Gaussian-gram gauss–krüger projection and the UTM projection can be approximated by x[utm]=0.9996 * x[Gauss],y[ utm]=0.9996 * y[Gauss], coordinate conversion (note: If the coordinate axis is shifted westward by 500000 meters, the conversion must be converted with the Y value minus 500000 multiplied by the scale factor plus 500000). From the point of view of the band, the beginning of the difference between the two, Gauss-gram Gauss–krüger projection from 0 degrees from the meridian every 6 degrees from the west to the east, the 1th with the central longitude of the 3°;utm projection from the longitude of the west of the 6 degrees from the western east of the belt, the 1th band of the central longitude of -177°, So the 1th band of the Gaussian-gram Gauss–krüger projection is the 31st band of UTM. In addition, the East pseudo-offset of the two projections is 500 km, the Gaussian-gram Gauss–krüger projection North Pseudo-offset is zero, the UTM Northern hemisphere projection North pseudo-offset is zero, the southern hemisphere is 10000 km.

Therefore, in the measurement, mapping and data conversion, we should pay attention to the difference between the two, to avoid errors, resulting in unnecessary errors.

3.3 Meaning of the coordinate system in ArcGIS

There are two kinds of geographic coordinate system (geographic coordinate system) and projected coordinate system (projected coordinate system) in ArcGIS. The former defines which geographic coordinate system to use, while the latter defines both the geographic and projected coordinate systems.

The meanings of each of these parameters are as follows:

4. Seven parameter conversion

After understanding the above sections, let's explore the issues that are frequently encountered in the project. The first problem is the conversion between the two ellipsoid, such as: WGS84 's latitude-longitude coordinates are converted to XIAN80 latitude and longitude coordinates.

4.1 Method of conversion and acquisition of parameters

The seven-parameter method (including Bursa model, one-step model, Helmant, etc.) is a more rigorous and common method to solve this problem. The seven parameters involved are: X-shift, Y-shift, Z-shift, X-rotation, y-rotation, Z-rotation, scale-change K. These seven parameters can be obtained by selecting more than 3 conversion control points in the area where conversion is required.

If the range is not large, the distance between the farthest point is not greater than 30Km (XP), this can be used with three parameters (Molodenski model), that is, x translation, y translation, z translation, while the x rotation, y rotation, z rotation, scale change K is considered 0. So the three parameters are just a special case of the seven parameter. The three parameters can be obtained with only 1 control points.

4.2 Explain the meaning of each parameter in detail

A. Three coordinate translation (x, y, Z), which is the coordinate difference between the coordinates origin of the two space coordinate system;

B. Three axes of rotation (α, β, γ)), the XYZ axes of two spatial right-angled coordinate systems can be coincident by specifying an angle by rotating three axes in order.

C. Scale factor K, which is the ratio of the length of the same line in two space coordinate systems, to achieve scale conversion. Usually the K value is almost equal to 1.

4.3 note

The seven-parameter conversion is for spatial direct coordinates that convert the spatial coordinates of a geographic coordinate system to another coordinate system.

The origin of the spatial Cartesian coordinate system is located at the center of the Earth's reference ellipsoid, the z-axis is parallel to the Earth's rotation axis and points to the reference ellipsoid, the x-axis is to the Prime meridian of the reference ellipsoid, and the y-axis and the x-axis and z-axis form a right-hand The geodetic coordinate system is established on the basis of the geodetic datum, and the geodetic datum is based on the reference ellipsoid, and the geodetic coordinate system is called the ellipsoid coordinate system.

5. Four parameter conversion 5.1 conversion method and four parameter acquisition

Converting between planar coordinates in a different coordinate system of an ellipsoid uses a planar transformation. At present, it is generally divided into four parameters and planar mesh fitting two methods, with four parameter method in domestic use more.

The mathematical implication of the four parameters is that the equation of the dependent variable (y) changes with the independent variable (x) by means of the equations containing four parameters.

For example, in Zhuhai, there are both the plane coordinates of Beijing 54 and the plane coordinates of Zhuhai, and the conversion between the two coordinates uses four parameters. The four-parameter acquisition requires two control point pairs.

Of course, it is more accurate to provide mesh fit data and then mesh fit.

5.2 Mathematical significance of four-parameter conversion

Four-parameter mode is y= (a-d)/[1+ (x/c) ^b]+d

A: Asymptote valuation on the curve.

B: The slope of the curve.

C: The maximum combination of the corresponding dose of half.

D: Asymptote value under curve.

The formula for solving the equations of multiple equations by iterative or approximation method is:

y= (a-d)/(1+ (X/C) ^b) + D

6. Learn the conversion steps with examples

Example: In a surveying area of the Pearl River, a coordinate conversion of WGS-84 coordinates to the Pearl River coordinate system (54 ellipsoid) is required, and the entire conversion process is:

The specific process is:

A. Convert the latitude and longitude of the WGS84 to the coordinates of the WGS84 space Cartesian coordinate system.

B. Seven parameters are obtained using the control points between the local three WGS84 and the local coordinates (ellipsoid is BJ54).

C. Convert the spatial Cartesian coordinates of the WGS84 to the spatial cartesian coordinates of the BJ54 coordinate system using the seven parameters.

D. Convert the spatial coordinate of the BJ54 to the latitude and longitude coordinates of the BJ54.

E. The BJ54 coordinates of the latitude and longitude of this time are Gaussian Gauss–krüger projected into BJ54 plane coordinates.

F. Four parameters are obtained using the two control points between the plane coordinates of the local BJ54 and the local plane coordinates.

G. Convert BJ54 plane coordinates to local plane coordinates using four parameters.

7. Summary

Write this article to see a lot of information. Thanks to Jack Deng's http://www.cnblogs.com/tiandi/archive/2011/12/03/2274903.html. Thanks to Fang Qinglin's http://blog.sciencenet.cn/blog-586485-457129.html, thanks to the blog Park and the senior. Thanks to Oo Lwin's new geography system principle book.

In this paper, the Gauss G-Gauss–krüger projection of the positive and inverse algorithm, as well as seven parameters and four parameter method can be found on the Internet, here will not be posted to everyone.

A brief analysis of the seven-parameter conversion method and the four-parameter conversion method used in the project and the basic surveying knowledge involved