R^3/Projection image/Linear mapping/Example

In many situations, a certain object (like a cube) in space ${\displaystyle {}\mathbb {R} ^{3}}$ shall be drawn in the plane ${\displaystyle {}\mathbb {R} ^{2}}$. One possibility is to work with a certain projection. This is a linear mapping

${\displaystyle \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2}}$

that is given (with respect to the standard bases ${\displaystyle {}e_{1},e_{2},e_{3}}$ and ${\displaystyle {}f_{1},f_{2}}$) by

${\displaystyle e_{1}\longmapsto f_{1},\,e_{2}\longmapsto af_{1}+bf_{2},\,e_{3}\longmapsto f_{2},}$

where the coefficients ${\displaystyle {}a,b}$ are usually chosen in the range ${\displaystyle {}[{\frac {1}{3}},{\frac {1}{2}}]}$. Linearity has the effect that parallel lines are mapped to parallel lines (unless they are mapped to a point). The point ${\displaystyle {}(x,y,z)}$ is mapped to ${\displaystyle {}(x+ay,by+z)}$. The image of the object under such a linear mapping is called a projection image.