Transformations play a special role in photogrammetry, so learning the mathematical principles and their calculations helps us better understand and employ photogrammetry tools. A mapping performs the task of transferring a set of coordinates from a source coordinate system to a target coordinate system. In this section, we review the simplest and most practical 2D transformations used in photogrammetry and describe a simple way to numerically estimate their parameters.
1- 2D Conformal
The first transformation that we investigate is a 2D conformal transformation that aim to connect two cartesian coordinate system each in 2D. The formula for the conformal transformation is
where, [x;y] is the coordinates of a point in the first cartesian coordinate system, and [x’;y’] is the coordinates of the same point in the second 2D coordinate system,[x_0;y_0] is the shift between two co. sys. 
is the rotation angle, and 
is the scale between them.By simplifying those equations we get
A numerical calculation technique to obtain the unknowns of the above equation is the method of using subsidiary unknowns, in such a way that we rewrite the above equations as below.
Here we see that for the simplicity of calculations, functions of the main variables are defined as new variables. We call these new variables “secondary variables”. This technique makes the calculation process easier. Now we can write the equations for a set of corresponding points and solve the equations as follows
The system of equations is solved by employing the method of least square
Finally, a residual vector is calculated as
We should notice that after calculating the secondary unknown, the main unknowns could be easily calculated from them.
2-2D Affine Transformation
The second two-dimensional transformation that we examine in this section is 2D affine transformation, in which, in addition to the conformal parameters, other parameters such as the non-perpendicularity of the axes are also considered. The relation of affine transformation in the form of matrix multiplication is expressed as follows
By mixing scale parameters and matrix elements we have
For simplicity of calculations, similar to the conformal case, we define functions of the main variables as
Finally, we can solve these equations by knowing at least 3 points.
As we can see in above equations, the number of degrees of freedom in this case is less than conformal transformations. We use least-square technique to solve these equations.
When calculating the inverse, care about a possible singularity in the matrix of coefficients. Singularities arise especially when the points are located on a line, e.g.
3- 2D Projective transformation
Projective transformation is one of the most important two-dimensional transformations used in photogrammetry. Projective trasformation is generally used in two situations: 2D to 2D and 3D to 2D. One reason for the importance of projective transformations is that they are compatible with the conditions of projective geometry of regular cameras.
The formula for 2D-2D projective transformation is as follows
To solve these equations, one may rewrite them as
We can see that the above equations are linear with respect to the parameters, therefore, we can easily solve them by knowing at least 4 points.
3D to 2D projective transformation has 11 parameters and expressed as
The solution is similar to 2D-2D case. One of the most important applications of this transformation is in space resection with DLT method that we will see in other pages.
4- 2D Polynomial transformation
This transformation is defined by
The number of terms and their complexity completely depends on the application that we want to address. Here we have freedom to choose the number of terms and coefficients, therefore, a wider space for different models exists.
5- 3D Conformal transformation
This transformation is defined to connect two sets of 3D cartesian co. sys. The formula is
There exists 7 unknown parameters in this equation that are defined as main unknown. One of the most important applications of 3D conformal in photogrammetry is to solve absolute orientation problem. We will explain two methods to solve it in that section.
6- 3D Affine transform
This transformation is defined by helper variables as
Therefore, the system of observational equations is formed as
The solution will be similar to the previous sections using least-square technique.
7- 3D Projective transformation
A 3D projective transform that connects two sets of Cartesian 3D co.sys. is defined by employing the following 15 parameters
8- 3D polynomial Transformation
This is the last transformation that we list here. The equations are as follows
9-Choosing level of complexity for an optimum transformation
In choosing a transformation, we should always keep in mind that a transformation with a smaller number of parameters, as long as it leads to acceptable errors for a problem, is better than a more complex mapping with a larger number of parameters. Of course, this point does not mean that a simpler transformation is preferred in all cases. In many problems, a simple transformation may not be able to logically adapt to the geometry of the problem in question, so as a rule of thumb, it can be stated that the simplest mapping that leads to acceptable error margins is desirable.