Understanding and Applying Complex Transformations: Mapping a Circle Under Inversion
In the realm of complex analysis, transformations of geometric figures play a crucial role in understanding their properties and behaviors. One such transformation is the inversion z → 1/z, which is particularly interesting when applied to circles. This article will guide you through the process of mapping a circle z - 2i 1 under this transformation and introduces Mbius transformations as a broader class of complex mappings.
Parametrization and Transformation
The given circle z - 2i 1 can be parametrized using complex exponentials, as follows:
z(t) 2i e^{it}, where t in [0, 2π]
This parametrization simplifies the equation of the circle, centered at 2i, with a radius of 1 unit.
Applying the Transformation z → 1/z
Substitute the parametrization into the transformation w 1/z.
w(t) 1/(z(t)) 1/(2i e^{it})
Simplifying the Expression
To simplify w(t), multiply the numerator and denominator by the conjugate of the denominator.
w(t) 1/(2i e^{it}) cdot (2i - e^{-it})/(2i - e^{-it}) (2i - e^{-it})/(2i e^{it} - e^{-it})
The denominator simplifies as follows:
2i e^{it} - e^{-it} 2i e^{it} - e^{-it} 4 - 1 e^{it} - e^{-it} -4 1 -3
Thus, w(t) (2i - e^{-it}) / -3 - (2i/3) e^{-it}
Expressing w(t) in Standard Form
We can now rewrite the expression for w(t) as:
w(t) - (2/3)i (1/3) e^{-it}
Identifying the Image
The term (1/3) e^{-it} describes a circle of radius 1/3, centered at the point -2/3i. Thus, the image of the circle under the transformation is a circle of radius 1/3, centered at -2/3i.
Summary of the Mapping
The image of the circle is:
left| w - frac{2}{3}i right| frac{1}{3}
Additional Insights: Mbius Transformations
The mapping z → 1/z is a nice example of a complex transformation known as Mbius transformation. Mbius transformations are a class of transformations that encompass scalings, translations, and rotations. They are determined by four complex-valued parameters and can be represented by a 2 times 2 matrix.
Key Properties of Mbius Transformations
Lines and circles are mapped to lines and circles, including the Riemann sphere where lines are considered as special cases of circles.
The center of the new circle will be the image of the old center. For example, 2i maps to 1/(2i) -i/2.
The radius of the new circle can be computed from a point on the original circle. For instance, if 1/2i, then 1/(1/2i) (1-2i)/5, giving a radius of 1/sqrt{5}.
Understanding these mappings and properties provides a deeper insight into the behavior of complex numbers and their applications in various fields of mathematics and engineering.