Reflection (geometry)

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In Euclidean geometry, a reflection is a linear operation σ on ${\displaystyle {\mathbb{ℝ}}^3}$ with σ² = E, the identity map. This property of σ is called involution. An involutory operator is non-singular and σ⁻¹ = σ. Reflecting twice an arbitrary vector brings back the original vector :

${\displaystyle \sigma (\overrightarrow{\mathbf{𝐫}})=\overrightarrow{\mathbf{𝐫}}{}^{\prime }\text{and}\sigma (\overrightarrow{\mathbf{𝐫}}{}^{\prime })=\overrightarrow{\mathbf{𝐫}}.}$

The operation σ is an isometry of ${\displaystyle {\mathbb{ℝ}}^3}$ onto itself, which means that it preserves inner products and that its inverse is equal to its adjoint,

${\displaystyle {\sigma }^{\mathrm{T}}={\sigma }^{-1}(=\sigma ).}$

Hence reflection is also symmetric: σ^T = σ. From (det(σ))² = det(E) = 1 follows that isometries have determinant ±1. Those with positive determinant are rotations, while reflections have determinant −1. Because σ is symmetric it has real eigenvalues; since the extension of an isometry to a complex space is unitary, its (complex) eigenvalues have modulus 1. It follows that the eigenvalues of σ are ±1. The product of the eigenvalues being its determinant, −1, the sets of eigenvalues of σ are either {1, 1, −1}, or {−1, −1, −1}. An operator with the latter set of eigenvalues is equal to −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[edit]

If ${\displaystyle \widehat{\mathbf{𝐧}}}$ is a unit vector normal (perpendicular) to a plane—the mirror plane—then ${\displaystyle (\widehat{\mathbf{𝐧}}\cdot \overrightarrow{\mathbf{𝐫}})\widehat{\mathbf{𝐧}}}$ is the projection of ${\displaystyle \overrightarrow{\mathbf{𝐫}}}$ on this unit vector. From the figure it is evident that

${\displaystyle \overrightarrow{\mathbf{𝐫}}-\overrightarrow{\mathbf{𝐫}}{}^{\prime }=2(\widehat{\mathbf{𝐧}}\cdot \overrightarrow{\mathbf{𝐫}})\widehat{\mathbf{𝐧}}⟹\overrightarrow{\mathbf{𝐫}}{}^{\prime }=\overrightarrow{\mathbf{𝐫}}-2(\widehat{\mathbf{𝐧}}\cdot \overrightarrow{\mathbf{𝐫}})\widehat{\mathbf{𝐧}}}$

If a non-unit normal ${\displaystyle \overrightarrow{\mathbf{𝐧}}}$ is used then substitution of

${\displaystyle \widehat{\mathbf{𝐧}}=\frac{\overrightarrow{\mathbf{𝐧}}}{|\overrightarrow{\mathbf{𝐧}}|}\equiv \frac{\overrightarrow{\mathbf{𝐧}}}{n}}$

gives the mirror image,

${\displaystyle \overrightarrow{\mathbf{𝐫}}{}^{\prime }=\overrightarrow{\mathbf{𝐫}}-2\frac{(\overrightarrow{\mathbf{𝐧}}\cdot \overrightarrow{\mathbf{𝐫}})\overrightarrow{\mathbf{𝐧}}}{n^2}}$

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

${\displaystyle \overrightarrow{\mathbf{𝐫}}{}^{\prime }=[\mathbf{𝐄}-\frac{2}{n^2}\overrightarrow{\mathbf{𝐧}}\otimes \overrightarrow{\mathbf{𝐧}}]\overrightarrow{\mathbf{𝐫}},}$

where E is the 3×3 identity matrix.

Dyadic products satisfy the matrix multiplication rule

${\displaystyle [\overrightarrow{\mathbf{𝐚}}\otimes \overrightarrow{\mathbf{𝐛}}][\overrightarrow{\mathbf{𝐜}}\otimes \overrightarrow{\mathbf{𝐝}}]=(\overrightarrow{\mathbf{𝐛}}\cdot \overrightarrow{\mathbf{𝐜}})(\overrightarrow{\mathbf{𝐚}}\otimes \overrightarrow{\mathbf{𝐝}}).}$

By the use of this rule it is easily shown that

${\displaystyle {[\mathbf{𝐄}-\frac{2}{n^2}\overrightarrow{\mathbf{𝐧}}\otimes \overrightarrow{\mathbf{𝐧}}]}^2=\mathbf{𝐄},}$

which confirms that reflection is involutory.

Reflection in a plane not through the origin[edit]

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

${\displaystyle \overrightarrow{\mathbf{𝐫}}=\overrightarrow{\mathbf{𝐬}}-\overrightarrow{\mathbf{𝐭}},\overrightarrow{\mathbf{𝐫}}{}^{\prime }=\overrightarrow{\mathbf{𝐬}}{}^{\prime }-\overrightarrow{\mathbf{𝐭}}}$

Use of the equation derived earlier gives

${\displaystyle \overrightarrow{\mathbf{𝐬}}{}^{\prime }-\overrightarrow{\mathbf{𝐭}}=\overrightarrow{\mathbf{𝐬}}-\overrightarrow{\mathbf{𝐭}}-2(\widehat{\mathbf{𝐧}}\cdot (\overrightarrow{\mathbf{𝐬}}-\overrightarrow{\mathbf{𝐭}}))\widehat{\mathbf{𝐧}}.}$

And hence the equation for the reflected pair of vectors is,

${\displaystyle \overrightarrow{\mathbf{𝐬}}{}^{\prime }=\overrightarrow{\mathbf{𝐬}}-2(\widehat{\mathbf{𝐧}}\cdot (\overrightarrow{\mathbf{𝐬}}-\overrightarrow{\mathbf{𝐭}}))\widehat{\mathbf{𝐧}},}$

where ${\displaystyle \widehat{\mathbf{𝐧}}}$ is a unit vector normal to the plane. Obviously ${\displaystyle \overrightarrow{\mathbf{𝐭}}}$ and ${\displaystyle \widehat{\mathbf{𝐧}}}$ are proportional, they differ only by scaling. Therefore, the equation can be written solely in terms of ${\displaystyle \overrightarrow{\mathbf{𝐭}}}$,

${\displaystyle \overrightarrow{\mathbf{𝐬}}{}^{\prime }=\overrightarrow{\mathbf{𝐬}}-2\frac{\overrightarrow{\mathbf{𝐭}}\cdot (\overrightarrow{\mathbf{𝐬}}-\overrightarrow{\mathbf{𝐭}})}{t^2}\overrightarrow{\mathbf{𝐭}},t^2\equiv \overrightarrow{\mathbf{𝐭}}\cdot \overrightarrow{\mathbf{𝐭}}.}$

Two consecutive reflections[edit]

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 φ is equal to twice the angle between the mirror planes. Indeed, the angle ∠ AP'M = ∠ MP'B = α and ∠ BP'N = ∠ NP'C = β. The rotation angle ∠ AP'C ≡ φ = 2α + 2β and the angle between the planes is α+β = φ/2.

It is obvious that the product of two reflections is a rotation. Indeed, a reflection is an isometry and has determinant −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 ${\displaystyle \overrightarrow{\mathbf{𝐬}}}$ and of the second ${\displaystyle \overrightarrow{\mathbf{𝐭}}}$, then the rotation is represented by the matrix

${\displaystyle [\mathbf{𝐄}-\frac{2}{t^2}\overrightarrow{\mathbf{𝐭}}\otimes \overrightarrow{\mathbf{𝐭}}][\mathbf{𝐄}-\frac{2}{s^2}\overrightarrow{\mathbf{𝐬}}\otimes \overrightarrow{\mathbf{𝐬}}]=\mathbf{𝐄}-\frac{2}{t^2}\overrightarrow{\mathbf{𝐭}}\otimes \overrightarrow{\mathbf{𝐭}}-\frac{2}{s^2}\overrightarrow{\mathbf{𝐬}}\otimes \overrightarrow{\mathbf{𝐬}}+\frac{4}{t^2{s}^2}(\overrightarrow{\mathbf{𝐭}}\cdot \overrightarrow{\mathbf{𝐬}})(\overrightarrow{\mathbf{𝐭}}\otimes \overrightarrow{\mathbf{𝐬}})}$

The (i,j) element if this matrix is equal to

${\displaystyle {\delta }_{ij}-\frac{2t_i{t}_j}{t^2}-\frac{2s_i{s}_j}{s^2}+\frac{4t_i{s}_j(\sum _k{t}_k{s}_k)}{t^2{s}^2}.}$

This formula is used in vector rotation.