Twist extension

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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 :

Here is the scaling factor and the angle of rotation .

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

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 :

The coordinates describe a final shift . The fixed point of the figure can be determined from this by solving a linear system of equations .

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

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

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.

See also

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