# Equation-based methods

Equation-based transformation methods can be classified into the following four method types.

## Three-parameter methods

The simplest datum transformation method is a geocentric, or three-parameter, transformation. The geocentric transformation models the differences between two datums in the XYZ or 3D Cartesian coordinate system. One datum is defined with its center at 0,0,0. The center of the other datum is defined at some distance (dx,dy,dz or ΔX,ΔY,ΔZ) in meters away.

Usually the transformation parameters are defined as going "from" a local datum "to" World Geodetic System (WGS) 1984 or another geocentric datum.

The three parameters are linear shifts and are always in meters.

## Seven-parameter methods

A more complex and accurate datum transformation is possible by adding four more parameters to a geocentric transformation. The seven parameters are three linear shifts (dx,dy,dz), three angular rotations around each axis (rx,ry,rz), and a scale factor.

The rotation values are given in decimal seconds, while the scale factor is in parts per million (ppm). The rotation values are defined in two different ways: as positive either clockwise or counterclockwise as you look toward the origin of the XYZ systems.

The previous equation is how the United States and Australia define the equations and is called the Coordinate Frame rotation transformation. The rotations are positive counterclockwise. Europe uses a different convention called the Position Vector transformation. Both methods are sometimes referred to as the Bursa–Wolf method. In the Projection Engine, the Coordinate Frame and Bursa–Wolf methods are the same. Both Coordinate Frame and Position Vector methods are supported, and it is easy to convert transformation values from one method to the other simply by changing the signs of the three rotation values. For example, the parameters to convert from the WGS 1972 datum to the WGS 1984 datum with the Coordinate Frame method are (in the order dx,dy,dz,rx,ry,rz,s):

(0.0, 0.0, 4.5, 0.0, 0.0, -0.554, 0.227)

To use the same parameters with the Position Vector method, change the sign of the rotation so these are the new parameters:

(0.0, 0.0, 4.5, 0.0, 0.0, +0.554, 0.227)

It's impossible to tell from the parameters alone which convention is being used. If you use the wrong method, your results can return inaccurate coordinates. The only way to determine how the parameters are defined is by checking a control point whose coordinates are known in the two systems.

The Molodensky–Badekas method is a variation of the seven-parameter methods. It has an additional three parameters that define the XYZ origin of rotation. Sometimes this point is known as the origin of the datum, or geographic coordinate system. Given the XYZ origin of rotation point, it is possible to calculate an equivalent Coordinate Frame transformation. The dx, dy, and dz values will change, but the rotation and scale values will remain the same.

## Molodensky method

The Molodensky method converts directly between two geographic coordinate systems without actually converting to an XYZ system. The Molodensky method requires three shifts (dx,dy,dz) and the differences between the semimajor axes (Δa) and the flattenings (Δf) of the two spheroids. The Projection Engine automatically calculates the spheroid differences according to the datums involved.

- h = ellipsoid height (meters)
- Φ = latitude
- λ = longitude
- a = semimajor axis of the spheroid (meters)
- b = semiminor axis of the spheroid (meters)
- f = flattening of the spheroid
- e = eccentricity of the spheroid

M and N are the meridional and prime vertical radii of curvature, respectively, at a given latitude. The equations for M and N are:

You solve for Δλand ΔΦ. The amounts are added automatically by the Projection Engine.

## Abridged Molodensky method

The Abridged Molodensky method is a simplified version of the Molodensky method. See the equations below: