Geometric Transformation

A geometric transformation is a Linear Transformation that maps ${\mathbb{R}}^2\to {\mathbb{R}}^2$ or ${\mathbb{R}}^3\to {\mathbb{R}}^3$

Properties §

Visually, a linear transformation can be deduced in a Cartesian Plane if two properties are satisfied:

1. Every vector line in ${\mathbb{R}}^2$ maps to either another line or a vector (point): §

The equation of a line is given by:

Then a transformed vector line is given by:

$$T({\vec{r}}_0+t\times \vec{d})$$

And we can use the properties of a linear transformation, specifically that it preserves preserves linear combinations:

$$T({\vec{r}}_0+t\times \vec{d})=T({\vec{r}}_0)+tT(\vec{d})$$

Which gives another vector line with direction vector = $T(\vec{d})$ originating at $T({\vec{r}}_0)$. If $T(\vec{d})=\vec{0}$, then the line degenerates into a point.

2: Every parallelogram created by two vectors maps to another parallelogram (or it’s degenerate forms): §

A parallelogram here is defined by 2 linearly independent vectors and the zero vector of ${\mathbb{R}}^2$. Alternatively, we could use another vector instead of the zero vector, but I’m lazy like that:

Let $\vec{u},\vec{v}\in {\mathbb{R}}^2$. Then a parallelogram, $P$ can be defined with vertices.

$$P=(\vec{0},\vec{u},\vec{v},\vec{u}+\vec{v})$$

And it’s transformed version is:

$$P^{\prime }=(\vec{0},T(\vec{u}),T(\vec{v}),T(\vec{u})+T(\vec{v}))$$