In this explainer, we will learn how to translate and rotate a complex number in the complex plane.
When we consider transformation of the complex plane, we are thinking of functions that map a two-dimensional space to a two-dimensional space. On first impression, it can be difficult to understand how to think about and interpret such functions. The difficulty arises from the fact that the most common techniques we use to understand real functions tend to fall down when we try to understand functions mapping two dimensions to two dimensions. This explainer will introduce a number of techniques we use to understand complex functions.
We will begin by looking at some simple examples to introduce the techniques we use.
Example 1: Translations of the Complex Plane
Find an equation for the image of under the transformation of the complex plane .
Answer
We are considering a transformation from the -plane to another complex plane which we will call the -plane. We can then represent the transformation as
We can find an equation for the image of by first expressing in terms of as follows:
At this point, we can substitute this into the equation which yields
This represents a circle of radius 2 centered at the point represented by the complex number . We can represent both the -plane and the -plane as follows.
The previous example introduces us to one of the key techniques we use to understand transformations of the complex plane: considering the image of different curves under the transformation. Understanding the effect of a given transformation on circles and lines will often give us a reasonable impression of what the transformation is doing. More generally, looking at the effect of a transformation on curves and regions is a technique we use to help visualize more complex types of transformation.
Additionally, the previous example demonstrated that a transformation in the form , where is constant (sometimes described as the transformation ), represents a translation by the vector .
In the next example, we will consider transformations in the form where .
Example 2: Dilations of the Complex Plane
Find an equation for the image of under the transformation of the complex plane .
Answer
We are considering a transformation from the -plane to another complex plane which we will call the -plane. We can represent this transformation as
We can find an equation for the image of by first expressing in terms of as follows:
At this point, we can substitute this into the equation which yields
Using the properties of the modulus, we can rewrite this as which is equivalent to
This represents a circle of radius centered at the origin. We can represent both the -plane and the -plane as follows.
The previous example demonstrated that transformations of the form where is constant are dilations by a scale factor of . Although this is a true statement, strictly speaking, the previous example is not sufficient to demonstrate this. We saw that these transformations map circles centered at the origin to other circles centered at the origin with a scaled radius. However, it could be possible that the transformation has also rotated points in the plane. Although in this case this is not true, it does demonstrate that generally we should consider the images of both points and circles to ensure we have a better picture of what a transformation is doing.
Example 3: Rotations of the Complex Plane
Find an equation for the image of under the transformation that maps the -plane to -plane.
Answer
We can find an equation for the image of by first expressing in terms of as follows:
We can now substitute this into the equation which yields
Using the properties of the argument, we can rewrite this as
Since , we can simplify this to
This represents a half-line whose end point is at the origin, which makes an angle of with the positive real axis.
The previous example showed us that, under the transformation , the image half-line at the origin which makes an angle of to the real axis is a half-line at the origin which makes an angle of . This demonstrates that the transformation rotated the plane by . Moreover, we know that it did not scale the line since the modulus of is one.
Notice that had we considered the image of a circle in the previous example, we would have found that its image would be a circle in the -plane with the same center and radius. This would not have been very insightful. This is why we often need to consider the image of both lines and circles to gain a clearer understanding of what a given transformation is doing.
Combining the results of examples 2 and 3, we can see that multiplication by a general complex number can be understood as a combination of the transformation we get by multiplying by the real number , which is a dilation by scale factor , and the transformation we get by multiplying by , which is a counterclockwise rotation about the origin through an angle of .
Basic Transformation of the Complex Plane
Let where are constant.
- The transformation represents a translation by the vector .
- The transformation represents a dilation by scale factor and a counterclockwise rotation about the origin by .
More involved transformations can be built up by combining these transformations. For example, we can understand the transformation of the form as a composition of the transformations and . That is, we can understand it to be a dilation by scale factor with a counterclockwise rotation about the origin by , followed by a translation by the vector representing the complex number .
In the next couple of examples, we will consider composite transformation of this form.
Example 4: Compound Transformations of the Complex Plane
Find an equation for the image of the half-line under the transformation .
Answer
We are considering a transformation from the -plane to another complex plane which we will call the -plane. We can represent this transformation as
We can find an equation for the image of by first expressing in terms of as follows:
Substituting this into the equation yields
Expressing the subject of the argument as a single fraction, we have which we can simplify to
Using the properties of the argument, we can rewrite this as
Since , we have
This represents a half-line whose end point is at which makes an angle of with the positive horizontal. We can interpret this transformation as the combination of two transformations , a counterclockwise rotation about the origin by radians, and , a translation by the vector . We can represent this visually as follows.
Example 5: Composite Transformations of the Complex Plane
Find an equation for the image of under the transformation of -plane to the -plane given by .
Answer
We begin by expressing in terms of as follows:
We can now substitute this into the equation to find the equation (in terms of ) for the image of this locus under the given transformation. Hence,
Expressing the subject of the modulus as a single fraction, we have
Factoring out from the numerator, we have
We can now use the properties of the modulus to rewrite this as
Since , we can express this as
This is a circle of radius 6 centered at . We can interpret this transformation as the combination of three transformations: , a dilation centered at the origin with scale factor 2, followed by , a translation by the vector , followed by , a counterclockwise rotation about the origin by radians. We can represent this visually as follows.
In the final example, we will consider a transformation which, unlike the transformations we have considered thus far, does not map straight lines to straight lines, and consequently, it is more challenging to gain an understanding of the effect of the transformation. However, using the techniques we have learned we can consider its effect on lines and circles and, as a result, start to build up a picture of what the transformation is doing.
Example 6: Transformations of the Complex Plane
A transformation which maps the -plane to the -plane is given by .
- Find a Cartesian equation for the image of under the transformation.
- Find a Cartesian equation for the image of .
- Find a Cartesian equation for the image of .
Answer
For the points in the -plane, we will denote the real and imaginary parts as and respectively, whereas, for the plane, we will denote them and .
Part 1
We will first find a complex equation for the image of in the -plane; then we will transform it to a Cartesian equation in terms of and . We define . Taking the modulus of both sides of this equation, we have
Using the properties of the modulus, we can rewrite this as
Given the fact that we would like to find the image of , we can simply replace with 2 in the equation above which gives
Hence, the image of under the transformation is . Since this is a circle of radius 64 centered at the origin, we can give the Cartesian equation of this as
We can visualize this as follows.
Part 2
The points in the -plane which satisfy can all be written in the form . To consider the image of this under the transformation , we substitute this into the equation as follows:
Expanding the parentheses, we have
Since , we can equate the real and imaginary parts to get the two simultaneous equations
To get a Cartesian equation from these two equations, we would like to eliminate . From the second equation, we have . We can substitute this into the first equation to get
We can visualize this as follows.
Part 3
All of the points in the -plane which satisfy can be written in the form . Substituting this into the equation , we have
Expanding the parentheses, we have
Since , we can equate the real and imaginary parts to get the two simultaneous equations
To get a Cartesian equation form these two equations, we would like to eliminate . From the second equation, we have . We can substitute this into the first equation to get
We can visualize this as follows.
As we have seen in the previous example, understanding a transformationβs effect on lines and circles helps us gain an understanding of its overall effect on the complex plane, even in the case of more general transformations. There are many other techniques which are useful for gaining this understanding, for example, considering the preimage of lines and circles, color maps, vector fields, and discovering the fixed points of the transformation.
Key Points
- We can gain an understanding of transformations in the complex plane by considering their effect on lines and circles.
- For a complex number , where , the transformation represents a translation by the vector .
- The transformation represents a dilation by scale factor and a counterclockwise rotation about the origin by .
- A composite transformation can be built up by composing multiple transformations. We can understand their affect by considering the effect of each transformation in turn.
- To gain insight into more complex transformations such as , we can also consider how it maps lines and circles. However, understanding the exact nature of the transformation will generally be more challenging.
