4.2.3 Reflections/Mirroring

Now we want to describe the reflection $S_w$ on a plane orthogonal to some (unit) vector $w$. Since any vector $v$ can be written as

$$v=\langle v\vert w\rangle w+(v-\langle v\vert w\rangle w),$$

where $v-\langle v\vert w\rangle w$ is orthogonal to $w$ (use $\langle v-\langle v\vert w\rangle w,w\rangle=\langle v\vert w\rangle(1-\Vert w\Vert^2)=0$), we get

$$S_w(v) = S_w\Bigl(\langle v\vert w\rangle w\Bigr)+S_w\Bigl(v-\langle v\vert w\rangle w\Bigr)$$

$$= -\langle v\vert w\rangle w + \Bigl(v-\langle v\vert w\rangle w\Bigr) = v - 2\,\langle v\vert w\rangle w.$$

Figure: Reflection

In coordinates:

$$(v_1,v_2,v_3) \mapsto \Bigl(v_1 -2\Bigl(\sum_k v_k\,w_k\Bigr)\,w_1,v_2 -2\Bigl(\sum_k v_k\,w_k\Bigr)\,w_2, v_3 -2\Bigl(\sum_k v_k\,w_k\Bigr)\,w_3\Bigr)$$

$$= \begin{pmatrix}1-2w_1^2 & -2w_1\,w_2 & -2w_1\,w_3 \\ -2w_2\,w_1... ...2 & 1-2w_3^2 \end{pmatrix} \cdot \begin{pmatrix}v_1 \\ v_2 \\ v_3 \end{pmatrix}$$

$$= (\operatorname{id}- 2\,w\cdot w^t)\cdot v = v - 2\,w\,\langle w\vert v\rangle$$

These reflections are length preserving:

$$(\operatorname{id}- 2\,w\cdot w^t)^t\cdot (\operatorname{id}- 2\,w\cdot w^t) = (\operatorname{id}- 2\,w\cdot w^t)\cdot (\operatorname{id}- 2\,w\cdot w^t)$$

$$= \operatorname{id}- 2\,w\cdot w^t - 2\,w\cdot w^t + 4\,w\cdot w^t\cdot w\cdot w^t = \operatorname{id},$$

or, because of Pythagoras,

$$\Vert v-2\,\langle v\vert w\rangle w\Vert^2 =\Vert v-\langle v\vert w\rangle w\Vert^2+\Vert\pm\langle v,w\rangle w\Vert^2$$

$$=\Vert v-\langle v\vert w\rangle w+\langle v\vert w\rangle w\Vert^2 =\Vert v\Vert^2.$$

Note here, that a linear mapping given by multiplication with the matrix $A$ is length preserving, iff

$$v^t\cdot w=\langle v\vert w\rangle=\langle Av\vert Aw\rangle =(Av)^t\cdot Aw=v^t\,A^t\,A\,w$$

for all $v$ and $w$, i.e. iff $A^t\cdot A=\operatorname{id}$.

The composition of the two 2-dimensional reflections $\langle a,b\rangle\mapsto \langle -a,b\rangle$ and $\langle a,b\rangle\mapsto \langle a,-b\rangle$ amounts to mirroring $v\mapsto -v$ at the center $O$. The composition of two general reflections is a rotation:

$$(S_{w_2}\o S_{w_1})(v) = S_{w_1}(v) - \langle S_{w_1}(v)\vert w_2\rangle w_2$$

$$= v - \langle v\vert w_1\rangle w_1 - \langle v - \langle v\vert w_1\rangle w_1 \vert w_2\rangle w_2$$

$$= v - \langle v\vert w_1\rangle w_1 - \langle v\vert w_2\rangle w_2 + \langle v\vert w_1\rangle \langle w_1 \vert w_2\rangle w_2$$

This is a rotation around $w_1\times w_2$ by twice the angle between $w_1$ und $w_2$.

Figure: Rotation as two-fold reflection