Linear Transformation of a Complex Number
Suppose we have a set of complex points and we have two complex numbers and then we can obtain another set of complex points using 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 controls the scaling by scale of and the rotation through angle . For positive rotation angle we have
- Complex number translate the results of multiplication of into .
Suppose we have these set of complex points that produce a shape.
The plot of Real and Imaginary of the complex points is shown below as a complex plane.
We want to design a complex linear transformation that will rotate counterclockwise and scale it at 0.5 and translate it (0.5, 1)
We want to find the value of such that and . For the translation, we simply use .
We know that and .
Thus, we have
and . 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 . Thus, we have
We input the second equation to the first equation to give