By Kardi Teknomo, PhD .
ComplexNumber

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Suppose we have a set of complex points Complex Number Tutorial:Complex Linear Transformation and we have two complex numbers Complex Number Tutorial:Complex Linear Transformation and Complex Number Tutorial:Complex Linear Transformation then we can obtain another set of complex points Complex Number Tutorial:Complex Linear Transformation using linear transformation

Complex Number Tutorial:Complex Linear Transformation

This complex linear transformation is rigid body transformation which contains a rotation about the origin and a scaling and a translation.

Properties of Complex Linear Transformation

  • Complex number Complex Number Tutorial:Complex Linear Transformation controls the scaling by scale of Complex Number Tutorial:Complex Linear Transformation and the rotation through angle Complex Number Tutorial:Complex Linear Transformation . For positive rotation angle we have

Complex Number Tutorial:Complex Linear Transformation , Complex Number Tutorial:Complex Linear Transformation

  • Complex number Complex Number Tutorial:Complex Linear Transformation translate the results of multiplication of Complex Number Tutorial:Complex Linear Transformation into Complex Number Tutorial:Complex Linear Transformation .

Example:

Suppose we have these set of complex points that produce a shape.

Complex Number Tutorial:Complex Linear Transformation

The plot of Real and Imaginary of the complex points is shown below as a complex plane.

Complex Number Tutorial:Complex Linear Transformation

We want to design a complex linear transformation that will rotate Complex Number Tutorial:Complex Linear Transformation counterclockwise and scale it at 0.5 and translate it (0.5, 1)

Answer

We want to find the value of Complex Number Tutorial:Complex Linear Transformation such that Complex Number Tutorial:Complex Linear Transformation and Complex Number Tutorial:Complex Linear Transformation . For the translation, we simply use Complex Number Tutorial:Complex Linear Transformation .

We know that Complex Number Tutorial:Complex Linear Transformation and Complex Number Tutorial:Complex Linear Transformation .

Thus, we have

Complex Number Tutorial:Complex Linear Transformation and Complex Number Tutorial:Complex Linear Transformation . However, the range of function tan in computer is usually from -pi to +pi, thus, we need to adjust by shifting pi/2 to get the correct result.

From trigonometry, we know that Complex Number Tutorial:Complex Linear Transformation . Thus, we have

Complex Number Tutorial:Complex Linear Transformation and Complex Number Tutorial:Complex Linear Transformation

We input the second equation to the first equation to give

Complex Number Tutorial:Complex Linear Transformation

Complex Number Tutorial:Complex Linear Transformation

Complex Number Tutorial:Complex Linear Transformation --> Complex Number Tutorial:Complex Linear Transformation --> Complex Number Tutorial:Complex Linear Transformation

Thus, Complex Number Tutorial:Complex Linear Transformation

Complex Number Tutorial:Complex Linear Transformation

See Also : Fractal Geometry , Application of Complex Number

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