Twist extension

A rotation extension is a similarity map that can be represented as a combination of the two geometric operations rotation and extension .

In the 2D space (plane) it is characterized by 2 transformation parameters, with an additional parallel shift by 4 parameters.
In the 3D case not dealt with here , there are 4 or 7 parameters, see 7-parameter transformation .

Euclidean plane

Center in the origin

Every rotation (with the exception of identity ) has exactly one fixed point , also called the center . If this fixed point is in the origin of the coordinates , the rotation extension can be written as a matrix multiplication :

${\displaystyle \left({\begin{matrix}x\\y\end{matrix}}\right)\mapsto \left({\begin{matrix}r\cos \phi &-r\sin \phi \\r\sin \phi &r\cos \phi \end{matrix}}\right)\cdot \left({\begin{matrix}x\\y\end{matrix}}\right)=\left({\begin{matrix}r\cos \phi \cdot x-r\sin \phi \cdot y\\r\sin \phi \cdot x+r\cos \phi \cdot y\end{matrix}}\right)}$

Here is the scaling factor and the angle of rotation . ${\displaystyle r\neq 0}$ ${\displaystyle \phi }$

In the complex plane , the same mapping can be written as a complex multiplication:

${\displaystyle z\mapsto re^{i\phi }\cdot z\qquad {\text{mit }}z\in \mathbb {C} ,r\in \mathbb {R} \setminus \{0\},\phi \in \mathbb {R} }$

Any center

If the fixed point of the rotational extension is outside the origin, one must either carry out a translation of the coordinates or calculate with homogeneous coordinates :

${\displaystyle \left({\begin{matrix}x\\y\\1\end{matrix}}\right)\mapsto \left({\begin{matrix}r\cos \phi &-r\sin \phi &t_{x}\\r\sin \phi &r\cos \phi &t_{y}\\0&0&1\end{matrix}}\right)\cdot \left({\begin{matrix}x\\y\\1\end{matrix}}\right)=\left({\begin{matrix}r\cos \phi \cdot x-r\sin \phi \cdot y+t_{x}\\r\sin \phi \cdot x+r\cos \phi \cdot y+t_{y}\\1\end{matrix}}\right)}$

The coordinates describe a final shift . The fixed point of the figure can be determined from this by solving a linear system of equations . ${\displaystyle (t_{x},t_{y})}$

The general form of a rotational extension with any center can also be expressed in the complex plane:

${\displaystyle z\mapsto re^{i\phi }\cdot z+t\qquad {\text{mit }}z\in \mathbb {C} ,r\in \mathbb {R} \setminus \{0\},\phi \in \mathbb {R} ,t\in \mathbb {C} }$

In this form, the position of the center is particularly easy to find as a solution to the fixed point equation:

${\displaystyle z=re^{i\phi }\cdot z+t\quad \Rightarrow \quad z={\frac {t}{1-re^{i\phi }}}}$

For and , the formulas describe a parallel shift that is not counted as a twist because it cannot be made up of a twist and a twist. If, however, at the same time (or ), the formulas represent the identical figure , which counts as a special case to the rotational stretching, composed of a rotation by 0 ° and a stretching with the factor 1. ${\displaystyle r=1}$ ${\displaystyle \phi =0{\pmod {360^{\circ }}}}$ ${\displaystyle t_{x}=t_{y}=0}$ ${\displaystyle t=0}$

See also

• rotation
• Centric elongation
• Similarity transformation

Web links

• Interactive demonstration of a turning stretch