Click to follow **
1. Applicable to 4 parameters
Convert between two different 2D planar Cartesian coordinate systems, typically using a four-parameter model.
The four-parameter is suitable for the spatial coordinate conversion of a small-scale survey area, and the advantage over the seven-parameter conversion is that only two common known points are needed to convert it, and the operation is simple.
2. Four unknown parameters
There are four unknown parameters in this model, namely:
1) The translation of two coordinates (x, y), that is, the coordinate difference between the coordinate origins of the two planar coordinate systems.
2) The rotation angle of the plane coordinate axis a, by rotating an angle, the x and y axes of the two coordinate systems can be coincidental.
3) The scale factor k, that is, the ratio of the length of the same straight line in two coordinate systems, realizes the proportional transformation of the scale. Usually the k-value is almost equal to 1.
The mathematical meaning of the four-parameter is to use an equation with four parameters to express the law of the change of the dependent variable (y) with the independent variable (x).
It usually takes at least two common known points and four pairs of xy coordinates in two different planar Cartesian coordinate systems to deduce these four unknown parameters. After calculating these four parameters, we can convert the XY coordinate value of the next point in one plane Cartesian coordinate system into the XY coordinate value of another plane Cartesian coordinate system through the four-parameter equation system.
Three, four-parameter application
The four-parameter is very common in the use of GPS-RTK (as opposed to the seven-parameter argument), so let's calculate the four-parameter of GPS-RTK together.
1. The calculation process of four parameters
The four-parameter is a transformation parameter of two planar coordinate systems, with two translation values (δx, δy) and a rotation value (r) and a scale coefficient (m). As long as we have two coordinates of the plane coordinate system at our common point, we can solve the four parameters. Such a common point needs at least two, because one point can only establish two error equations, to solve four unknowns, at least two points need to establish four error equations, and if there are many points, use the least squares method.
Least Squares Concept:
The leastsquare method, also known as the leastsquare method, is a mathematical formula, mathematically called curve fitting, and the leastsquare method here refers specifically to linear regression equations. The formula is:
2. RTK calculates the internal process
Since the four parameters can only be carried out between the plane coordinate system, and the construction coordinate system is already the plane coordinates, we only need to convert the geodetic coordinates to the plane coordinates.
However, those who have used Gaussian projection calculus should know that Gaussian projection calculus requires ellipsoids and meridians, so we need to use WGS84 ellipsoids, because the geodetic coordinates provided by the design are based on WGS84.
*The meridian longitude cannot be changed, it can only be provided by design, the earth is an ellipsoid, **The meridian changes, and the Y value will be greatly deformed.
In this way, (bl) can be converted into (xy) by the Gaussian projection, and the calculation conditions of four parameters are met.