13.2 Representations of the Euclidean Plane
As we study transformational geometry of the plane we will use four distinct perspectives.
13.2.1 Synthetic Plane
The synthetic perspective follows the methods of Euclid without the use of coordinates or formulas. With this synthetic approach, the concept of distance and measurement is always with respect to a separately defined unit.
13.2.2 Cartesian Plane
In the 1600s René Descartes and Pierre de Fermat independently developed the use of a coordinate plane to provide an analytic foundation for the study of geometry. While some may consider the synthetic methods to be more axiomatic, we have shown in Chapter 4 that the construction of the real number system, and thus also \(\mathbb{R}^2\), is just as axiomatic using the axioms of set theory. When we follow the real analytic perspective of transformational geometry we will generally use the distance defined by the Euclidean distance, \[d\left( (x,y),(w,z)\right) = \sqrt{(w-x)^2+(z-y)^2},\] and we will consider the transformations as \(f:\mathbb{R}^2 \rightarrow \mathbb{R}^2\) with \[f(x,y)= \left( f_1(x,y), f_2(x,y)\right)\] where \(f_1\) and \(f_2\) are functions from \(\mathbb{R}^2\) to \(\mathbb{R}\).
Furthermore, in \(\mathbb{R}^2\) we see that lines can be expressed in the form \[\left\{ (x,y) \in \mathbb{R}^2 \: \vert \: y=mx+b \mbox{ for some } m,b\in \mathbb{R}\right\}\] and circles centered at the point \((x_0,y_0)\) with a radius of \(r\) can be expressed as \[\left\{ (x,y)\in \mathbb{R}^2 \vert (x-x_0)^2 + (y-y_0)^2 = r^2\right\}.\]
13.2.3 Vector Space
An alternative perspective of the plane involves viewing \(\mathbb{R}^2\) as a two-dimensional real vector space. From this perspective, we define \[\mathbb{R}^2 = \left\{ \langle x,y\rangle \: \vert \: x,y\in \mathbb{R}\right\}\] with the inner product defined as \[\langle x,y\rangle\cdot \langle w,z\rangle = xw+yz,\] and the norm of a vector defined as \[||\langle x,y\rangle ||=\sqrt{\langle x,y\rangle \cdot\langle x,y\rangle }=\sqrt{x^2+y^2}.\] We define vector addition defined by \(\langle x,y\rangle +\langle w,z\rangle = \langle x+w, y+z\rangle\), scalar multiplication by \(c \langle x,y\rangle =\langle cx,cy\rangle\), and so \(\langle x,y\rangle -\langle w,z\rangle = \langle x-w,y-z\rangle\) and we can define the distance between vectors as \[d\left( \langle x,y\rangle ,\langle w,z\rangle \right) = || \langle x,y\rangle -\langle w,z\rangle ||.\] We will focus much of our attention on linear transformations of this vector space represented by matrices, \[T_A\left(\langle x,y\rangle \right) = A\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax+by \\ cx+dy \end{pmatrix} = \langle ax+by, cx+dy\rangle .\]
13.2.4 Complex Plane
In the 1700s Leonard Euler proposed the concept of considering the complex numbers as isomorphic to the real plane, \[\mathbb{C} = \left\{ x+iy \: \vert \: (x,y)\in \mathbb{R}^2\right\}\] with distance \[d(w,z) = |w-z|.\] Then transformations of the plane are simply functions of the form \(f:\mathbb{C} \rightarrow \mathbb{C}\). The circle centered at the complex number \(c\) with radius \(r\) in \(\mathbb{C}\) can be expressed as \[C_{(c,r)} = \left\{ z\in \mathbb{C} \: \vert \: |z-c|=r\right\}.\] The details of the complex plane perspective are in Section 4.6.
13.2.5 Isometries
Recall that the definition of an isometry is a surjection that maintains distances between any two points. We will see that the uses of the different representations of the plane will help us to better understand and categorize these isometries.
Related Content Standards
- (HSG.CO.2) Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
The representation of the plane using complex numbers contains a significant amount of information. Using the properties of the complex numbers we can determine which functions \(f:\mathbb{C}\rightarrow \mathbb{C}\) are isometries.
Theorem 13.7 \(f:\mathbb{C} \rightarrow \mathbb{C}\) is an isometry if and only if \[f(z)= az+b \quad \mbox{or} \quad f(z)=a \overline{z} +b\] for some \(a,b\in \mathbb{C}\) with \(|a|=1\).
Proof. If \(f\) is an isometry, then \[|f(z)-f(w)|=|z-w|\] for all \(z,w\in \mathbb{C}\). In order to better understand these isometries we will first consider some simpler isometries.
Assume that \(g:\mathbb{C}\rightarrow \mathbb{C}\) is an isometry such that \(g(0)=0\) and \(g(1)=1\). Then the properties of isometries implies that for all complex numbers \(z\), \[|g(z)|=|z| \quad \mbox{and} \quad |g(z)-1|=|z-1|.\] Squaring each of these equations gives us the equivalent equations \[g(z) \overline{g(z)} = z \overline{z} \quad \mbox{and} \quad (g(z)-1)\overline{(g(z)-1)} = (z-1)\overline{(z-1)}.\] Using properties of complex conjugation and distribution the second equation can be rewritten as \[g(z)\overline{g(z)} - (g(z)+\overline{g(z)}) +1 = z\overline{z} - (z+\overline{z}) +1\] and substituting the first equation we have that \[g(z)+\overline{g(z)} = z+\overline{z}.\] This means that \(\mbox{Re}(g(z)) = \mbox{Re}(z)\). Since \(|g(z)|=|z|\) we can infer that \(\mbox{Im}(g(z))= \pm \mbox{Im}(z)\) and so \(g(z)=z\) or \(g(z)=\overline{z}\).
If \(f:\mathbb{C}\rightarrow \mathbb{C}\) is a generic isometry, we can let \(b=f(0)\) and see that \(h:\mathbb{C}\rightarrow\mathbb{C}\) defined by \(h(z)=f(z)-b\) is an isometry such that \(h(0)=0\). This would then imply that \(|h(1)| = |h(1)-h(0)|=|1|\). We can then let \(a=h(1)\) and we see that \(|a|=1\) and so \(a^{-1}=\overline{a}\). If we let \(g:\mathbb{C}\rightarrow \mathbb{C}\) be defined by \(g(z)=\overline{a} h(z) = \overline{a} (f(z)-b)\) we see that \(g\) is an isometry with \(g(0)=0\) and \(g(1)=1\). So by the argument in the previous paragraph, \(g(z)=z\) or \(g(z)=\overline{z}\). Therefore, \[f(z)= az+b \quad \mbox{or} \quad f(z)=a \overline{z} +b\] for some \(a,b\in \mathbb{C}\) with \(|a|=1\).
Over the next several sections we will see how the properties of these expressions correspond to the transformations of translations, rotations, reflections, and glide-reflections.
Definition 13.9 Let \(I\) be the set of isometries of the plane. \[I = \left\{ f: \mathbb{C} \rightarrow \mathbb{C} \:\vert \: f(z)=a z+b \mbox{ or } f(z)=a \bar{z}+b \mbox{ for some } a, b \in \mathbb{C} \mbox{ with } |a|=1 \right\}\]
13.2.6 Exercises
For each of the following functions, \(f:\mathbb{R}^2 \rightarrow \mathbb{R}^2\), determine if \(f\) is an isometry. Justify your answer.
- \(f(x,y)= (x+2,y-3)\)
- \(f(x,y)=(x^2,y)\)
- \(f(x,y)=(-x,-y)\)
- \(\displaystyle{f(x,y)=\left( \frac{x+\sqrt{3}y}{2}, \frac{-\sqrt{3}x+y}{2}\right) }\)
- \(f(x,y)=(x,-y)\)
If \(T_A:\mathbb{R}^2\rightarrow \mathbb{R}^2\) is an isometry represented by a \(2\times 2\) matrix, \(A\), as \[T_A\left(\langle x,y\rangle \right) = A\begin{pmatrix} x \\ y \end{pmatrix},\] what can we say about the determinant of \(A\)?
For each of the following functions, \(f:\mathbb{C} \rightarrow \mathbb{C}\), determine if \(f\) is an isometry. Justify your answer by comparing \(|f(z)-f(w)|\) and \(|z-w|\) for generic complex numbers \(z\) and \(w\).
- \(f(z)=z+2-3i\)
- \(f(z)=z^2\)
- \(f(z) = \overline{z}+2i\)
- \(f(z)= 3 \overline{z}\)
- \(\displaystyle{f(z)= e^{i\frac{\pi}{4}} z +i}\)
Use the form of isometries for the complex plane, \[I = \left\{ f: \mathbb{C} \rightarrow \mathbb{C} \:\vert \: f(z)=a z+b \mbox{ or } f(z)=a \overline{z}+b \mbox{ for some } a, b \in \mathbb{C} \mbox{ with } |a|=1 \right\},\] to prove each of the theorems in the previous section about isometries.
- If \(f\) and \(g\) are isometries, then \(g\circ f\) is an isometry.
- If \(f\) is an isometry, then \(f^{-1}\) is invertible and \(f^{-1}\) is also an isometry.
- Isometries map lines to lines and circles to circles.
Use the form of isometries for the complex plane, \[I = \left\{ f: \mathbb{C} \rightarrow \mathbb{C} \:\vert \: f(z)=a z+b \mbox{ or } f(z)=a \overline{z}+b \mbox{ for some } a, b \in \mathbb{C} \mbox{ with } |a|=1 \right\},\] to show that \(I\), together with the operation of function composition, is a group.