Use of homogeneous coordinates
in 2D:
\(m = \underbrace{ \left[ \begin{array}{c} x \\ y \end{array} \right] }_{Euclidean ~coordinates} \Rightarrow \tilde{m} = \underbrace{ \left[ \begin{array}{c} x \\ y \\ 1 \end{array} \right] }_{Homogeneous ~coordinates}\)
in 3D:
\(M = \underbrace{ \left[ \begin{array}{c} X \\ Y \\ Z \end{array} \right] }_{Euclidean ~coordinates} \Rightarrow \tilde{M} = \underbrace{ \left[ \begin{array}{c} X \\ Y \\ Z \\ 1 \end{array} \right] }_{Homogeneous ~coordinates}\)
There are several advantages to that. For instance, we will see in the “Transformation between the camera reference frame and the sensor reference frame (retinal plane)” section that it enables expressing the pinhole model with a linear relation.