Create Transformation Relationship

1ppm = 0.001‰ is defined as one part per million (1 PPM)

Supported conversion relationship:

• Four parameters:
  • Translation parameter Dx (meter, default value: 0).
  • Translation parameter Dy (meter, default value: 0).
  • Rotation parameter Theta (second, default value: 0).
  • Zoom ratio t (default value: 1.0).
  • Elevation fitting: You can check whether to carry out elevation fitting. If elevation fitting is checked, you need to select predefined elevation fitting parameters from the drop-down list.

    Note: When converting with Dx -0, Dy -0, Theta = 0, t = 1.0, the original paint cloud will be obtained.

The four-parameter formula can be written as follows:

It can be solved according to the following formula:

• Seven parameters: The seven parameters in the Bursa model.
  • Translation parameter Dx (meter, default value: 0).
  • Translation parameter Dy (meter, default value: 0).
  • Translation parameter Dz (meter, default value: 0).
  • Rotation parameter Rx (meter, default value: 0).
  • Rotation parameter Ry (meter, default value: 0).
  • Rotation parameter Rz (meter, default value: 0).
  • Zoom ratio m (ppm).

    Note: The scale m in the seven parameters is different from the scale t in the four parameters. The two can be transformed into each other. The conversion relationship is m = (t - 1.0) 1e6. Assuming t is 0.9999988, then m = (0.9999988-1.0) 1e6 = -1.2.

The formula of Bursa's seven parameters can be written as follows:

The seven-parameter model can be converted into 3D affine transformation parameters, and the conversion relationship is:

Dx = Dx;

Dy = Dy;

Dz = Dz;

Rx(″) = Rx(″);

Rx(″) = Rx(″);

Rx(″) = Rx(″);

Mx(ppm) = m(ppm);

My(ppm) = m(ppm);

Mz(ppm) = m(ppm);

• 3D affine transformation:

• Dx, Dy, Dz (m, default value is "0"): Values (panning values) added to X, Y, Z.
• Mx, My, Mz (ppm, default value is "0"): Scale factors applied to X, Y, and Z. It has the same definition as the scale factor m in the seven parameters.
• Rx, Ry, Rz (second, default value is "0"): The angle of rotation about the X, Y, and Z axes in degrees.

  Tips: the number of digits after The decimal point must be at least 8 in oder to reach mm-level accuracy.

• Four parameters (displayed as two-dimensional transformation parameters): Two-dimensional transformation parameters, also known as XYMultiply.

Two-dimensional transformation parameters can be expressed in the following formula:

Among them, Px, Py, Pz, a, b, c, d, e are the transformation parameters, Sx, Sy, Sz are the original coordinates, and X, Y, Z are the transformed coordinates. This is often used as a 2D Helmert transformation. The parameters can be converted from four parameters. Assuming that the four parameters are: Dx, Dy, theta, t, the conversion method is:

Px = Dx;

Px = Dy;

Pz = 0.0;

Define k = t;

Define tRad = theta / 3600.0 / 180 * PI;

Then:

a = k * cos(tRad);

b = -k * sin(tRad);

c = k * sin(tRad);

d = k * cos(tRad);

e = 1.0;

• Linear transformation: The linear transformation is used to pan and zoom a point cloud. A panning parameter and a scaling parameter need to be set to the X, Y, Z coordinates, respectively.

Linear transformation is calculated using the following formula:

Among them: S_x, S_y. S_z are the scaling factors for the x, y, and z coordinates. P_x, P_y, and P_z are the panning parameters for the x, y, and z coordinates. x, y, and z are the original coordinates, and X, Y, and Z are coordinates obtained after the linear transformation.

• Elevation fitting parameter: It is used to calculate elevation anomalies when xy is known, and then correct Z.
  • Plane fitting. The plane fitting formula is ζ = a₀ + a₁x + a₂y; It contains three parameters a₀-a₂.
  • Linear interpolation. The linear interpolation formula is ζ = a₀ + a₁x + a₂x² + a₃x³; It contains four parameters a₀-a₃.
  • Conicoid Fitting. ζ = a₀ + a₁x + a₂y + a₃xy + a₄x² + a₅y²; It contains six parameters a₀-a₅.
  • The formula for the Fitting of the Cubic Bézier Curve and Surface is ζ = a₀ + a₁x + a₂y + a₃xy + a₄x² + a₅y² + a₆x²y + a₇xy² + a₈x³ + a₉y³; It contains ten parameters a₀-a₉.