How to calculate "real" parameters from affine transformation

Introduction

This page discusses how to compute physically significant parameters from an arbitrary linear transformation. The reverse process, calculating the affine parameters from the physically significant parameters, is articulated on another wiki page. The model used to describe how the raster maps onto real world coordinates is illustrated in the following figure:

This model contains all of the parameters except for translation. Translation may be added in after the coordinates have been scaled, rotated and sheared. The above illustration contains the physically significant parameters which will be calculated by this method. These are: θ_i, θ_ij, and the pixel size along the i_b and j_b basis vectors.

The inputs to this method are the four non-translation (non-offset) parameters of a linear transform. These are shown below:

Although these parameters have "common names" within the GIS community, the names are both lengthy and misleading. Within the context of this page, the coefficients o₁₁, o₁₂, o₂₁, and o₂₂ will be used.

This page is divided into sections. Each section describes the computation of one physically significant parameter.

Pixel size in the i direction

To calculate the pixel size in the i direction, the i unit vector is projected using the given transform. This gives the basis vector i_b, the magnitude of which is the pixel size. The basis vector is expressed as follows:

The pixel size is then calculated as follows:

Pixel size in the j direction

The pixel size in the j direction is computed in a manner similar to the pixel size in the i direction. The only difference is that the j unit vector is used in place of the i unit vector. This gives the basis vector j_b, the magnitude of which is the pixel size. The basis vector is expressed as follows:

The pixel size is then calculated as follows:

Rotation

The grid is rotated by the angle θ_i. This is the angle between the x axis of the reference frame and the i_b basis vector. The angle θ_i is considered positive in the clockwise direction, for consistency with compass headings. The calculation of θ_i is a two-step process: first the magnitude is calculated, then the sign is determined. Both steps involve using the ​dot product to determine the angle between θ_i and one of the axes of the target coordinate system (either the x axis or the y axis. The equations in this section refer to the angles and vectors defined in the following figure:

The first step is to calculate the magnitude of the angle between i_b and the x axis. This is the magnitude of θ_i.

The angle θ_i is defined as the angle from the x axis to i_b. It is positive in the clockwise direction. In the situation described in the above figure, this means that θ_i is negative if i_b is above the x axis, and positive if below. We determine whether i_b is above or below the x axis by finding the angle between i_b and the y axis. If i_b and the y axis are separated by less than 90 degrees, i_b is above the x axis, otherwise it is below.

So, if θ_test is less than 90, θ_i = - abs(θ_i). Otherwise, θ_i = abs(θ_i).

Basis vector separation angle

In this section, the method to calculate θ_ij is presented. This is similar to the method for the calculation of θ_i, but it is accomplished with respect to the rotated reference frame of i_b and i_bp instead of the x and y axes. The figure which represents this setup is as follows:

The first step is to calculate the magnitude of θ_ij, the angle between i_b and j_b.

Next, we need to determine the sign of θ_ij in a manner similar to how the sign for θ_i was determined. The angle θ_ij always represents the angle from i_b to j_b, and is positive counterclockwise for consistency with a right-handed coordinate system. To do this, we first need to calculate i_bp, which is perpendicular to i_b and forms a ​right-hand coordinate system with i_b. Observe that i_bp is i_b after a 90 degree counterclockwise rotation.

Now we can determine the size of the angle between j_b and i_bp. In this situation, any angle less than 90 degrees means that j_b is on the same side of i_b as i_bp. An angle more than 90 degrees means it lies on the opposite side.

So, if θ_test is more than 90 degrees, θ_ij = - abs(θ_ij). Otherwise, θ_ij = abs(θ_ij).

If θ_ij has any value other than +-90 degrees, the basis vectors are not orthogonal and the transformed pixels are diamond shaped. Probably the most common value for θ_ij is -90 degrees, which indicates that the transform "flips" the j axis.

Summary

The method presented here performs calculations in the coordinate system used by the geocoordinates. All of the parameters calculated by this method are tied to the same SRID referenced by the transform O. It is assumed that the x and y axes of the geospatial coordinate system are orthogonal.

See also

• Wikipedia article on the ​dot product.
• Wikipedia article on ​coordinate systems.
• Wikipedia article on the ​world file.
• How to calculate a transform based on physically significant parameters.