Question

Show that an orthogonal map of the plane is either a reflection, or a rotation.

Answer 1

Answer on Question #61931 – Math – Linear Algebra

Question

Show that an orthogonal map of the plane is either a reflection, or a rotation.

Solution

If $T \colon R^2 \to R^2$ is an orthogonal linear operator, then the standard matrix for $T$ can be expressed in the form

$$R_{\theta} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$$

($R_{\theta}$ rotates vectors by $\theta$ radians, counterclockwise)

or

$$H_{\frac{\theta}{2}} = \begin{bmatrix} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{bmatrix}.$$

The determinant can be used to distinguish between the two cases:

$$\det R_{\theta} = \cos^2 \theta + \sin^2 \theta = 1$$ $$\det H_{\frac{\theta}{2}} = -(\cos^2 \theta + \sin^2 \theta) = -1.$$

Thus, a $2 \times 2$ orthogonal matrix represents a rotation about the origin if $\det = 1$ and a reflection about a line through the origin if $\det = -1$. This sentence may be a definition of an orthogonal map of plane. If the definition of $2 \times 2$ matrix is $A^T A = I$, where $I$ is an identity matrix, then the determinant of an orthogonal matrix is equal to 1 or $-1$, because using properties of determinants and equality $A^T A = I$, obtain $\det(A) = \det(A^T)$ and

$$1 = \det(I) = \det(A^T A) = \det(A^T) \det(A) = (\det(A))^2, \text{ hence } \det(A) = 1 \text{ or } \det(A) = -1.$$

Lemma. Every orthogonal transformation of the Euclidean plane is a reflection or a rotation.

We consider the effect of an orthogonal transformation $T$ on the two vectors $\alpha = (1,0)$ and

$\beta = (0,1)$. The orthogonality of $T$ forces $T(\alpha)$ and $T(\beta)$ also to be of unit length and orthogonal to each other. Let $\theta$ and $\hat{\theta}$ be the counter-clockwise angles formed by $T(\alpha)$ and $T(\beta)$ with the $x$-axis. Then $\hat{\theta} = \theta \pm \frac{\pi}{2}$. In the case $\hat{\theta} = \theta + \frac{\pi}{2}$, $T$ is a rotation through angle $\theta$. In the case $\hat{\theta} = \theta - \frac{\pi}{2}$, $T$ is a reflection through the dashed line forming an angle of $\frac{\theta}{2}$ with the $x$-axis. In the pictures we have

www.AssignmentExpert.com