R^3/Projection image/Linear mapping/Example

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

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

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

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

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