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Section3.1Basic Transformations of \(\mathbb{C}\)

We begin with a definition.

Definition3.1.1

Given two sets \(A\) and \(B\text{,}\) a function \(f:A \to B\) is called one-to-one (or 1-1) if whenever \(a_1 \neq a_2\) in \(A\text{,}\) then \(f(a_1) \neq f(a_2)\) in \(B\text{.}\) The function \(f\) is called onto if for any \(b\) in \(B\) there exists an element \(a\) in \(A\) such that \(f(a) = b\text{.}\) A transformation on a set \(A\) is a function \(T: A \to A\) that is one-to-one and onto.

Following are two schematics of functions. In the first case, \(f: A \to B\) is onto, but not one-to-one. In the second case, \(g: A \to B\) is one-to-one, but not onto.

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A transformation \(T\) of \(A\) has an inverse function, \(T^{-1}\text{,}\) characterized by the property that the compositions \(T^{-1}\circ T(a) = a\) and \(T\circ T^{-1}(a) = a\) for all \(a\) in \(A\text{.}\) The inverse function \(T^{-1}\) is itself a transformation of \(A\) and it “undoes” \(T\) in this sense: For elements \(z\) and \(w\) in \(A\text{,}\) \(T^{-1}(w)=z\) if and only if \(T(z) = w\text{.}\)

In this section we develop the following basic transformations of the plane, as well as some of their important features.

Basic Transformations of \(\mathbb{C}\)

  • General linear transformation: \(T(z)=az+b\text{,}\) where \(a,b\) are in \(\mathbb{C}\) with \(a \neq 0\text{.}\)
  • Special cases of general linear transformations:

    • Translation by \(b\text{:}\) \(T_b(z) = z + b\)
    • Rotation by \(\theta\) about \(0\text{:}\) \(R_\theta(z)=e^{i\theta}z\)
    • Rotation by \(\theta\) about \(z_0\text{:}\) \(R(z)=e^{i\theta}(z-z_0)+z_0\)
    • Dilation by factor \(k \gt 0\text{:}\) \(T(z)=kz\)

  • Reflection across a line \(L\text{:}\) \(r_L(z) = e^{i\theta}\overline{z}+b\text{,}\) where \(b\) is in \(\mathbb{C}\text{,}\) and \(\theta\) is in \(\mathbb{R}\text{.}\)

Example3.1.2Translation

Consider the fixed complex number \(b\text{,}\) and define the function \(T_b: \mathbb{C} \to \mathbb{C}\) by \begin{equation*} T_b(z) = z + b. \end{equation*}

The notation helps us remember that \(z\) is the variable, and \(b\) is a complex constant. We will prove that \(T_b\) is a transformation, but this fact can also be understood by visualizing the function. Each point in the plane gets moved by the vector \(b\text{.}\) So, for starters, the origin, 0, has image \(T_b(0) = b\text{,}\) and every other point in the plane gets moved the same amount and in the same direction. So, two different points (like \(v\) and \(w\) in Figure 3.1.3) cannot get moved to the same image point (thus, the function is one-to-one). Also, any point in the plane is the image of some other point (just follow the vector \(-b\) to find this “pre-image” point), so the function is onto as well.

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Figure3.1.3Translation by \(b\text{.}\)

We now offer a formal argument that the translation \(T_b\) is a transformation. Recall, \(b\) is a fixed complex number.

That \(T_b\) is onto:

To show that \(T_b\) is onto, let \(w\) denote an arbitrary element of \(\mathbb{C}\text{.}\) We must find a complex number \(z\) such that \(T_b(z) = w\text{.}\) Let \(z = w - b\text{.}\) Then \(T_b(z) = z + b = (w - b) + b = w\text{,}\) and \(T\) is onto.

That \(T_b\) is one-to-one:

To show that \(T_b\) is 1-1 we must show that if \(z_1 \neq z_2\) then \(T_b(z_1) \neq T_b(z_2)\text{.}\) We do so by proving the contrapositive. Recall, the contrapositive of a statement of the form “If P is true then Q is true” is “If Q is false then P is false.” These statements are logically equivalent, which means we may prove one by proving the other. So, in the present case, the contrapositive of “If \(z_1 \neq z_2\) then \(T_b(z_1) \neq T_b(z_2)\)” is “If \(T_b(z_1) = T_b(z_2)\text{,}\) then \(z_1 = z_2\text{.}\)”. We now prove this statement.

Suppose \(z_1\) and \(z_2\) are two complex numbers such that \(T_b(z_1) = T_b(z_2)\text{.}\) Then \(z_1 + b = z_2 + b\text{.}\) Subtracting \(b\) from both sides we see that \(z_1 = z_2\text{,}\) and this completes the proof.

Example3.1.4Rotation about the origin

Let \(\theta\) be an angle, and define \(\displaystyle R_\theta: \mathbb{C} \to \mathbb{C}\) by \(\displaystyle R_\theta(z) = e^{i\theta}z.\)

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Figure3.1.5Rotation about 0 by \(\theta\text{.}\)

This transformation causes points in the plane to rotate about the origin by the angle \(\theta\text{.}\) (If \(\theta > 0\) the rotation is counterclockwise, and if \(\theta \lt 0\) the rotation is clockwise.) To see this is the case, suppose \(z = re^{i\beta}\text{,}\) and notice that \begin{equation*} R_\theta(z) = e^{i\theta}re^{i\beta} = re^{i(\theta+\beta)}. \end{equation*}

Example3.1.6Rotation about any point

To achieve a rotation by angle \(\theta\) about a general point \(z_0\text{,}\) send points in the plane on a three-leg journey: First, translate the plane so that the center of rotation, \(z_0\text{,}\) goes to the origin. The translation that does the trick is \(T_{-z_0}\text{.}\) Then rotate each point by \(\theta\) about the origin (\(R_{\theta}\)). Then translate every point back (\(T_{z_0})\text{.}\) This sequence of transformations has the desired effect and can be denoted by the composition \begin{align*} T(z) \amp = T_{z_0} \circ R_{\theta} \circ T_{-z_0}(z)\\ \amp = T_{z_0}(R_{\theta}(z - z_0))\\ \amp = T_{z_0}(e^{i\theta}(z - z_0))\\ \amp = e^{i \theta}(z - z_0) + z_0. \end{align*}

That the composition of these three transformations is itself a transformation follows from the next theorem.

Example3.1.8Dilation

The transformation \(T(z) = kz\) where \(k > 0\) is a real number is called a dilation; such a map stretches or shrinks points in the plane along rays emanating from the origin.

Indeed, if \(z = x + y i\text{,}\) then \(T(z) = k x + k y i\text{,}\) and \(z\) and \(T(z)\) are on the same line through the origin. If \(k \gt 1\) then \(T\) stretches points away from the origin. If \(0 \lt k \lt 1\text{,}\) then \(T\) shrinks points toward the origin. In either case, such a map is called a dilation.

Given complex constants \(a, b\) with \(a \neq 0\) the map \(T(z) = az + b\) is called a general linear tranformation. We show in the following example that such a map is indeed a transformation of \(\mathbb{C}\text{.}\)

Example3.1.9General Linear Transformations

Consider the general linear transformation \(T(z)=az+b\text{,}\) where \(a, b\) are in \(\mathbb{C}\) and \(a \neq 0\text{.}\) We show \(T\) is a transformation of \(\mathbb{C}\text{.}\)

That \(T\) is onto:

Let \(w\) denote an arbitrary element of \(\mathbb{C}\text{.}\) We must find a complex number \(z\) such that \(T(z) = w\text{.}\) To find this \(z\text{,}\) we solve \(w = az + b\) for \(z\text{.}\) So, \(z = \frac{1}{a}(w - b)\) should work (since \(a \neq 0\text{,}\) \(z\) is a complex number). Indeed, \(T(\frac{1}{a}(w - b)) = a \cdot \big[\frac{1}{a}(w - b)\big] + b = w.\) Thus, \(T\) is onto.

That \(T\) is one-to-one:

To show that \(T\) is 1-1 we show that if \(T(z_1) = T(z_2)\text{,}\) then \(z_1 = z_2\text{.}\)

If \(z_1\) and \(z_2\) are two complex numbers such that \(T(z_1) = T(z_2)\text{,}\) then \(a z_1 + b = a z_2 + b\text{.}\) By subtracting \(b\) from both sides we see that \(a z_1 = a z_2\text{,}\) and then dividing both sides by \(a\) (which we can do since \(a \neq 0\)), we see that \(z_1 = z_2\text{.}\) Thus, \(T\) is 1-1 as well as onto, and we have proved \(T\) is a transformation.

Note that dilations, rotations, and translations are all special types of general linear transformations.

We will often need to figure out how a transformation moves a collection of points such as a triangle or a disk. As such, it is useful to introduce the following notation, which uses the standard convention in set theory that \(a \in A\) means the element \(a\) is a member of the set \(A\text{.}\)

Definition3.1.10

Suppose \(T: A \to A\) is a transformation and \(D\) is a subset of \(A\text{.}\) The image of \(D\text{,}\) denoted \(T(D)\text{,}\) consists of all points \(T(x)\) such that \(x \in D\text{.}\) In other words, \begin{equation*} T(D) = \{a \in A ~|~ a = T(x) ~\text{for some}~ x \in D\}. \end{equation*}

For instance, if \(L\) is a line and \(T_b\) is translation by \(b\text{,}\) then it is reasonable to expect that \(T(L)\) is also a line. If one translates a line in the plane, it ought to keep its linear shape. In fact, lines are preserved under any general linear transformation, as are circles.

Example3.1.12The image of a disk

The image of the disk \(D = \{z \in \mathbb{C}~|~|z-2i| \leq 1\}\) under the transformation \(T: \mathbb{C} \to \mathbb{C}\) given by \(T(z) = 2z + (4-i)\) is the disk \(T(D)\) pictured in Figure 3.1.13.

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Figure3.1.13The image of the disk \(D\) under the transformation \(T(z) = 2z + (4-i).\)

We will be interested in working with transformations that preserve angles between smooth curves. A planar curve is a function \(\vec{r}:[a,b] \to \mathbb{C}\) mapping an interval of real numbers into the plane. A curve is smooth if its derivative exists and is nonzero at every point. Suppose \(\vec{r}_1\) and \(\vec{r}_2\) are two smooth curves in \(\mathbb{C}\) that intersect at a point. The angle between the curves measured from \(\vec{r}_1\) to \(\vec{r}_2\text{,}\) which we denote by \(\angle(\vec{r}_1,\vec{r}_2)\text{,}\) is defined to be the angle between the tangent lines at the point of intersection.

Definition3.1.14

A transformation \(T\) of \(\mathbb{C}\) preserves angles at point \(z_0\) if \(\angle(\vec{r}_1,\vec{r}_2) =\angle(T(\vec{r}_1),T(\vec{r}_2))\) for all smooth curves \(\vec{r}_1\) and \(\vec{r}_2\) that intersect at \(z_0\text{.}\) A transformation \(T\) of \(\mathbb{C}\) preserves angles if it preserves angles at all points in \(\mathbb{C}\text{.}\) A transformation \(T\) of \(\mathbb{C}\) preserves angle magnitudes if, at any point in \(\mathbb{C}\text{,}\) \(\angle(\vec{r}_1,\vec{r}_2) =|\angle(T(\vec{r}_1),T(\vec{r}_2))|\) for all smooth curves \(\vec{r}_1\) and \(\vec{r}_2\) intersecting at the point.

Definition3.1.16

A fixed point of a transformation \(T: A \to A\) is an element \(a\) in the set \(A\) such that \(T(a) = a\text{.}\)

If \(b \neq 0\text{,}\) the translation \(T_b\) of \(\mathbb{C}\) has no fixed points. Rotations of \(\mathbb{C}\) and dilations of \(\mathbb{C}\) have a single fixed point, and the general linear transformation \(T(z) = az + b\) has one fixed point as long as \(a \neq 1\text{.}\) To find this fixed point, solve \begin{equation*} z = az + b \end{equation*} for \(z\) to get the fixed point \(z = \frac{b}{1-a}\text{.}\) For instance, the fixed point of the transformation \(T(z) = 2z + (4-i)\) of Example 3.1.12 is found by solving \(z = 2z + 4 - i\text{.}\) That is, the map \(T\) in this example has fixed point \(z_o=-4 + i\text{.}\) While the map \(T(z) = 2z + (4-i)\) moves the disk \(D\) to the disk \(T(D)\text{,}\) the point \(z_o\) gets to hunker where it is and watch.

Definition3.1.17

A Euclidean isometry is a transformation \(T\) of \(\mathbb{C}\) with the feature that \(|T(z)-T(w)| = |z-w|\) for any points \(z\) and \(w\) in \(\mathbb{C}\text{.}\) That is, a Euclidean isometry preserves the Euclidean distance between any two points.

Example3.1.18Rotations and translations are Euclidean isometries of \(\mathbb{C}\)

It is perhaps clear that translations, which move each point in the plane by the same amount in the same direction, ought to be isometries. Rotations are also isometries. In fact, the general linear transformation \(T(z)=az+b\) will be a Euclidean isometry so long as \(|a|=1\text{.}\) Indeed, \begin{align*} |T(z)-T(w)| \amp = |az + b - (aw + b)|\\ \amp = |a(z-w)|\\ \amp = |a| |z-w|\\ \amp = |z - w|. \tag{since $|a|=1$} \end{align*} Translations and rotations about a point in \(\mathbb{C}\) are general linear transformations of this type, so they are also Euclidean isometries.

Example3.1.19Reflection about a line

Reflection about a line \(L\) is the transformation of \(\mathbb{C}\) defined as follows: Each point on \(L\) gets sent to itself, and if \(z\) is not on \(L\text{,}\) it gets sent to the point \(z^*\) such that line \(L\) is the perpendicular bisector of segment \(zz^*\text{.}\)

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Reflection about \(L\) is defined algebraically as follows. If \(L\) happens to be the real axis then \begin{equation*} r_L(z) = \overline{z}. \end{equation*}

For any other line \(L\) we may arrive at a formula for reflection by rotating and/or translating the line to the real axis, then taking the conjugate, and then reversing the rotation and/or translation.

For instance, to describe reflection about the line \(y = x + 5\text{,}\) first translate vertically by \(-5i\text{,}\) then rotate by \(-\frac{\pi}{4}\text{,}\) reflect about the real axis, rotate by \(\frac{\pi}{4}\text{,}\) and translate by \(5i\) to get the composition \begin{align*} z \amp \mapsto z - 5i\\ \amp \mapsto e^{-\frac{\pi}{4}i}(z - 5i)\\ \amp \mapsto \overline{e^{-\frac{\pi}{4}i}(z - 5i)}= e^{\frac{\pi}{4}i}(\overline{z} + 5i)\\ \amp \mapsto e^{\frac{\pi}{4}i}\cdot e^{\frac{\pi}{4}i}(\overline{z} + 5i)=e^{\frac{\pi}{2}i}(\overline{z}+5i)\\ \amp \mapsto e^{\frac{\pi}{2}i}(\overline{z}+5i) + 5i \end{align*}

Simplifying (and noting that \(e^{\frac{\pi}{2}i} = i\)), the reflection about the line \(L: y = x + 5\) has formula \begin{equation*} r_L(z) = i\overline{z} - 5 + 5i. \end{equation*}

In general, reflection across any line \(L\) in \(\mathbb{C}\) will have the form \begin{equation*} r_L(z) = e^{i\theta}\overline{z}+b \end{equation*} for some angle \(\theta\) and some complex constant \(b\text{.}\)

Reflections are more basic transformations than rotations and translations in that the latter are simply careful compositions of reflections.

We list some elementary features of reflections in the following theorem. We do not prove them here but encourage you to work through the details. We will focus our efforts in the following section on proving analogous features for inversion transformations, which are reflections about circles.

In fact, one can show that any Euclidean isometry can be expressed as the composition of at most three reflections. See, for instance, Stillwell [10] for a proof of this fact.

SubsectionExercises

1

Is \(T(z) = -z\) a translation, dilation, rotation, or none of the above?

2

Show that the general linear transformation \(T(z) = a z + b\text{,}\) where \(a\) and \(b\) are complex constants, is the composition of a rotation, followed by a dilation, followed by a translation. Hint: View the complex constant \(a\) in polar form.

3

Prove that a general linear transformation maps circles to circles.

4

Suppose \(T\) is a rotation by 30\(^\circ\) about the point 2, and \(S\) is a rotation by \(45^\circ\) about the point 4. What is \(T\circ S\text{?}\) Can you describe this transformation geometrically?

5

Suppose \(T(z) = iz + 3\) and \(S(z) = -iz + 2\text{.}\) Find \(T\circ S\text{.}\) What type of transformation is this?

6

Find a formula for a transformation of \(\mathbb{C}\) that maps the open disk \(D = \{z ~|~ |z| \lt 2\}\) to the open disk \(D^\prime = \{ z ~|~ |z - i| \lt 5\}\text{.}\) Is this transformation unique, or can you think of two different ones that work?

7

Find a formula for reflection about the vertical line \(x = k\text{.}\)

8

Find a formula for reflection about the horizontal line \(y = k\text{.}\)

9

Find a formula for reflection in the plane about the line \(y = mx + b\text{,}\) where \(m \neq 0\text{.}\) Hint: Think about what angle this line makes with the positive \(x\)-axis.

10

Prove that the construction in Figure 3.1.21(b) determines the desired rotation.

11

\(S(z) = kz\) is a dilation about the origin. Find an equation for a dilation of \(\mathbb{C}\) by factor \(k\) about an arbitrary point \(z_0\) in \(\mathbb{C}\text{.}\)