# 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 :

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

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:

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

### 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 :

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

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:

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

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}$