Gaussian plane Cartesian coordinate system

The earth coordinate system is the basic Coordinate System of the Earth measurement. It is often used for the computation of earth problems, studying the shape and size of the Earth, preparing maps, positioning and operations for rocket and satellite launch and military aspects, it is inconvenient to directly use it for project construction planning, design, and construction. Therefore, it is necessary to convert the Earth coordinates on the sphere to the plane according to certain mathematical rules, that is, to use the theory of map projection to draw topographic maps for planning and construction.

An Elliptical decent is a surface that cannot be expanded directly. Therefore, the elements of an elliptical decent are projected onto a plane according to certain conditions, which will always produce deformation. The projection method is usually selected based on the requirement that the Projection deformation does not affect the project. There are three types of map projection: Equal-angle projection, equal-area projection, and arbitrary projection.
The equisigned projection is also called the positive projection. It ensures that the differential image on the elliptical decent will be similar after being projected onto the plane. This is the basic requirement for topographic maps. Normal projection has two basic conditions:
① Corner preserving condition, that is, the angle size after projection remains unchanged.
② The length deformation is fixed, that is, the length will be deformed after projection, but at one point, the deformation ratio of the differential line segment in each direction is a constant k:
Formula: The length after ds-projection and the length on the dS-sphere.

1. Concept of Gaussian projection

Gauss is an outstanding German mathematician and surveyor. He proposed a horizontal elliptical cylindrical projection as a positive projection. It sets a horizontal elliptical column on the Earth's elliptical body, as shown in:

The center O of the elliptical body is located on the center axis of the Elliptical Cylinder. The north and south poles of the elliptical body are tangent to the Elliptical Cylinder, and a meridian is tangent to the Elliptical Cylinder. This meridian is called a central Meridian. Then, point and line on the elliptical decent surface are projected to the Elliptical Cylinder according to the positive projection conditions, and then cut along the N and S bus of the Elliptical Cylinder and generate a plane, that is, the Gaussian projection plane. On this plane:

① The Central meridian is a straight line, and its length is not deformed. The other meridian that leaves the central meridian is an arc, and the concave is directed to the central Meridian. The farther away from the central meridian, the larger the deformation. ② After projection, the equator is a straight line, and the equator is orthogonal to the central Meridian. ③ The weft line that leaves the equator is an arc and convex to the equator.
Gaussian projection can change the elliptical sphere into a plane, but the longer the deformation is from the central meridian, the larger the deformation will affect the ing and construction accuracy. In order to control the length deformation, the method of limiting the projection width is adopted in the measurement, that is, the projection area is restricted in the long and narrow area near the two sides of the central Meridian. This method is called band-based projection. The projection band width is divided by the longitude difference between two adjacent Meridian. There are different projection methods such as 6 ° band and 3 ° band. The 6° belt projection starts from the Greenwich Meridian in the UK and is projected from west to east at intervals of 6°. In this way, the elliptical sphere is divided into 60 bands numbered 1 ~ See Figure 60:

The longitude (L) of each central meridian can be calculated in the following formula:
In the formula, n is the belt number with 6 °.
If you know the longitude of a point, you can use the following formula to calculate the belt number of the point:
When there is an remainder, It is the integer quotient + 1 of n. The 3 ° belt is divided based on the 6 ° belt. Its central meridian is overlapped with the 6 ° belt central meridian in the odd band, with a total of 120 bands at intervals of 3 °, the central meridian longitude (L) of each region is:
Type. China has a vast territory and 11 6 ° belts, from 13 ~ 23 strip (Central meridian from 75 ° ~ 135 °), 21 3 ° bands, from 25 ~ 45. Beijing is located in 20th of the 6° belt, and the central meridian longitude is 117 °.2. Gaussian Cartesian coordinate system

Based on the characteristics of Gaussian projection, the point of intersection between the equator and the central meridian is the coordinate origin ., The central meridian direction is the X axis, and the northern direction is positive. The equator projection line is the Y axis, and the east direction is positive. Quadrants are arranged clockwise, as shown in:

The y value in the same projection band has positive and negative values. This is inconvenient for computing and use. In order to make y values positive, the ordinate axis is shifted to the West by 500 km, and the belt number is placed before the y coordinate, for example, in the 20th belt, to the west of the central meridian point P:
Gaussian Cartesian coordinates in the 20-belt are:
The Gaussian Cartesian coordinate system is different from the Cartesian coordinate system in mathematics, as shown in:

The X-axis and Y-axis are the ordinate coordinates of the Gaussian Cartesian coordinate system. The coordinate quadrants are divided clockwise into four quadrants. The angle starts from the northern direction of the X axis and is calculated clockwise. These definitions are different from those in mathematics. This approach is to directly use the triangle and analytic geometric formula in mathematics to calculate the measurement.