Reflection (geometry)

In Euclidean geometry, a reflection is a linear operation &sigma; on ℝ3 with the property &sigma;2 = E, the identity map. This property of &sigma; is called involution. An involution is non-singular and is equal to its inverse: &sigma;&minus;1 = &sigma;. Reflecting twice an arbitrary vector brings back the original vector:

\sigma( \vec{\mathbf{r}}\,) = \vec{\mathbf{r'}} \quad\hbox{and}\quad \sigma( \vec{\mathbf{r'}}) = \sigma^2({\vec\mathbf{r}}\,) = E(\vec{\mathbf{r}}) = \vec{\mathbf{r}}. $$ The operation &sigma; is an isometry of ℝ3 onto itself, which means that it preserves inner products and hence that its inverse is equal to its adjoint,

\sigma^\mathrm{T} = \sigma^{-1}\; ( = \sigma). \, $$ It follows that reflection is symmetric: &sigma;T = &sigma;. From the properties of determinants

\det(\sigma)^2 = \det(\sigma)\det(\sigma^\mathrm{T}) = \det(\sigma)\det(\sigma^{-1}) = \det(\sigma\sigma^{-1}) = \det{E} = 1\, $$ follows that isometries have det(&sigma;) = &plusmn;1. Those with determinant +1 are rotations; those with determinant &minus;1 are reflections.

A reflection &sigma; on ℝ3 has two sets of eigenvalues:  {1, 1, &minus;1} and {&minus;1, &minus;1,  &minus;1}. This follows because the eigenvalues of &sigma;2 = E are +1 and hence  the eigenvalues of &sigma; are &plusmn;1. The product of the eigenvalues being the determinant &minus;1, the sets of eigenvalues of &sigma; are either {1, 1, &minus;1}, or {&minus;1, &minus;1, &minus;1}. An operator with the latter set of eigenvalues is equal to &minus;E, minus the identity operator. This operator is known alternatively as inversion, reflection in a point, or parity operator. An operator with the former set of eigenvalues is reflection in a plane. Reflections in a plane are the subject of this article.

Sometimes one finds the concept of "reflections in a line", these are rotations over 180°, see rotation matrix.

Reflection in a plane
If $$\hat{\mathbf{n}}$$ is a unit vector normal (perpendicular) to a plane&mdash;the mirror plane&mdash;then $$ (\hat{\mathbf{n}}\cdot\vec{\mathbf{r}})\hat{\mathbf{n}}$$ is the projection of $$\vec{\mathbf{r}}$$ on this unit vector. From the figure it is evident that

\vec{\mathbf{r}} - \vec{\mathbf{r}}\,' = 2 (\hat{\mathbf{n}}\cdot\vec{\mathbf{r}})\, \hat{\mathbf{n}} \;\Longrightarrow\; \vec{\mathbf{r}}\,' = \vec{\mathbf{r}} - 2 (\hat{\mathbf{n}}\cdot\vec{\mathbf{r}})\hat{\mathbf{n}} $$ If a non-unit normal $$\vec{\mathbf{n}}$$ is used then substitution of

\hat{\mathbf{n}} = \frac{\vec{\mathbf{n}}}{ |\vec{\mathbf{n}}|} \equiv \frac{\vec{\mathbf{n}}}{n} $$ gives the mirror image,

\vec{\mathbf{r}}\,' = \vec{\mathbf{r}} - 2 \frac{ (\vec{\mathbf{n}}\cdot\vec{\mathbf{r}})\vec{\mathbf{n}}}{n^2} $$

Sometimes it is convenient to write this as a matrix equation. Introducing the dyadic product, we obtain

\vec{\mathbf{r}}\,' = \left[ \mathbf{E} - \frac{2}{n^2} \vec{\mathbf{n}} \otimes\vec{\mathbf{n}} \right] \; \vec{\mathbf{r}}, $$ where E is the 3&times;3 identity matrix.

Dyadic products satisfy the matrix multiplication rule

[\vec{\mathbf{a}}\otimes\vec{\mathbf{b}}]\, [ \vec{\mathbf{c}}\otimes\vec{\mathbf{d}}] = (\vec{\mathbf{b}} \cdot \vec{\mathbf{c}}) \big( \vec{\mathbf{a}}\otimes\vec{\mathbf{d}} \big). $$ By the use of this rule it is easily shown that

\left[ \mathbf{E} - \frac{2}{n^2} \vec{\mathbf{n}} \otimes\vec{\mathbf{n}} \right]^2 = \mathbf{E}, $$ which confirms that reflection is involutory.

Reflection in a plane not through the origin
In Figure 2 a plane, not containing the origin O, is considered that is  orthogonal to the vector $$\vec{\mathbf{t}}$$. The length of this vector is the distance from O to the plane. From Figure 2, we find

\vec{\mathbf{r}} = \vec{\mathbf{s}} - \vec{\mathbf{t}}, \quad \vec{\mathbf{r}}\,' = \vec{\mathbf{s}}\,' - \vec{\mathbf{t}} $$ Use of the equation derived earlier gives

\vec{\mathbf{s}}\,' - \vec{\mathbf{t}} = \vec{\mathbf{s}} - \vec{\mathbf{t}} - 2 \big(\hat{\mathbf{n}}\cdot (\vec{\mathbf{s}} - \vec{\mathbf{t}})\big)\hat{\mathbf{n}}. $$ And hence the equation for the reflected pair of vectors is,

\vec{\mathbf{s}}\,' = \vec{\mathbf{s}} - 2 \big(\hat{\mathbf{n}}\cdot (\vec{\mathbf{s}} - \vec{\mathbf{t}})\big)\hat{\mathbf{n}}, $$ where $$\hat{\mathbf{n}}$$ is a unit vector normal to the plane. Obviously $$\vec{\mathbf{t}}$$ and $$\hat{\mathbf{n}}$$ are proportional, they differ only by scaling. Therefore, the equation can be written solely in terms of $$\vec{\mathbf{t}}$$,

\vec{\mathbf{s}}\,' = \vec{\mathbf{s}} - 2 \frac{\vec{\mathbf{t}}\cdot (\vec{\mathbf{s}} - \vec{\mathbf{t}})}{t^2}\vec{\mathbf{t}}, \quad t^2 \equiv \vec{\mathbf{t}}\cdot \vec{\mathbf{t}}. $$

Two consecutive reflections
Two consecutive reflections in two intersecting planes give a rotation around the line of intersection. This is shown in Figure 3, where PQ is the line of intersection. The drawing on the left shows that reflection of point A in the plane through PMQ brings the point A to B. A consecutive reflection in the plane through PNQ brings B to the final position C. In the right-hand drawing it is shown that the rotation angle  &phi;  is equal to twice the angle between the mirror planes. Indeed, the angle &ang; AP'M = &ang; MP'B = &alpha; and &ang; BP'N = &ang; NP'C = &beta;. The rotation angle &ang; AP'C &equiv; &phi; = 2&alpha; + 2&beta; and the angle between the planes is &alpha;+&beta; = &phi;/2.

It is obvious that the product of two reflections is a rotation. Indeed, a reflection is  an isometry and has determinant &minus;1. The product of two isometric operators is again an isometry and the rule for determinants is det(AB) = det(A)det(B), so that the product of two reflections is an isometry  with unit determinant, i.e., a rotation.

Let the normal of the first plane be $$\vec{\mathbf{s}}$$ and of the second $$\vec{\mathbf{t}}$$, then the rotation is represented by the matrix

\left[ \mathbf{E} - \frac{2}{t^2} \vec{\mathbf{t}} \otimes\vec{\mathbf{t}} \right]\, \left[ \mathbf{E}  - \frac{2}{s^2} \vec{\mathbf{s}} \otimes\vec{\mathbf{s}} \right] = \mathbf{E} - \frac{2}{t^2} \vec{\mathbf{t}} \otimes\vec{\mathbf{t}} - \frac{2}{s^2} \vec{\mathbf{s}} \otimes\vec{\mathbf{s}} + \frac{4}{t^2 s^2} (\vec{\mathbf{t}}\cdot\vec{\mathbf{s}})\; \big(\vec{\mathbf{t}} \otimes\vec{\mathbf{s}}\big) $$ The (i,j) element if this matrix is equal to

\delta_{ij} - \frac{2 t_i t_j }{t^2} - \frac{2 s_i s_j }{s^2} + \frac{4 t_i s_j (\sum_k t_k s_k)}{t^2 s^2}. $$ This formula is used in vector rotation.