In coplanarity equations, the goal is to obtain a simple and understandable mathematical form between 1- the coordinates of the photo points in two stereo images and 2- the relative orientation parameters of the stereo pair. To begin with, we consider the following stereo pair
In this stereo pair, the ground point could be seen by both images. If we consider the three-dimensional image coordinate system of the first image as a local coordinate system, then the rotations and displacements of the first image will be equal to zero
In this case, the orientation and relative transitions of the second image compared to the first will be
Using the above notation, any point in the 3D coordinate system of the second image could be transferred to the 3D coordinate system of the first image with the following relationship.
The above relation is a three-dimensional conformal transformation, where the scale is unit. Now, suppose that we denote (xy_1)_1, (xy_2)_2 as three-dimensional coordinates of the image points in 3D coordinate system of images, where (xy_1)_1 is an image point in the first image and in the three-dimensional coordinate system of the first image and (xy_2)_2 is projection in the second image (in 3D coordinate system of the second image).
The coplanar condition could be expressed such that three vectors (xy_1)_1, (xy_2)_1 and (X_02)_1 are coplanar. From geometry we know that if three vectors are coplanar, then the following statement applies to them
In the above formula, the symbol “x” is external multiplication of two vectors, and [B]_x is the matrix form of the external multiplication that was explained earlier. If we write this relation for the three given vectors, we will have
In vector form we have
Now by converting vector operations into matrix multiplications we have
In the above formula, pay attention that we have used the symbol (.)^T for the transpose of a matrix. In this regard, we reached our desired result and were able to establish a relationship between the image coordinates and relative orientation parameters. We consider the product of two middle matrices as Essential matrix
Now we can rewrite the previous equation as follows according to the definition of the Essential matrix
The above equation is a basic formula that includes 1- unknown parameters of relative orientation and 2- observed image points. Note that this relationship is in the form of an equation for each pair of image points, so to solve the 5 unknown parameters of the relative orientation, at least five corresponding stereo points (that corresponds to 5 model points) are required.