Reflection (geometry)

In analytic geometry, a reflection is a linear operation σ on $\mathbb {R} ^{3}$ with σ² = 1, i.e., σ is an involution, and σ⁻¹ = σ. Reflecting twice an arbitrary vector brings back the original vector :

$\sigma ({\vec {\mathbf {r} }}\,)={\vec {\mathbf {r} }}\,'\quad {\hbox{and}}\quad \sigma ({\vec {\mathbf {r} }}\,'\,)={\vec {\mathbf {r} }}.$

The operation σ is orthogonal, i.e., preserves inner products, so that

$\sigma ^{\mathrm {T} }=\sigma ^{-1}=\sigma .\,$

Hence reflection is orthogonal and symmetric. Further it has determinant −1. Because σ is symmetric it has real eigenvalues; since it is orthogonal its eigenvalues have modulus 1. In other words its eigenvalues are ±1. The product of the eigenvalues being the determinant (−1), the eigenvalues of σ are either {1, 1, −1}, or {−1, −1, −1}. An operator with the latter set of eigenvalues is minus the identity operator, this operator is known alternatively as inversion, reflection in a point, or parity operator. An operator with the first set of eigenvalues is reflection in a plane. Reflection in a plane will be considered further in this article.

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Fig. 1. The vector $\scriptstyle {\vec {\mathbf {r} }}$ goes to $\scriptstyle {\vec {\mathbf {r} }}'$ under reflection in a plane. The unit vector $\scriptstyle {\hat {\mathbf {n} }}$ is normal to mirror plane.

Reflection in a plane

If ${\hat {\mathbf {n} }}$ is a unit vector normal (perpendicular) to a plane—the mirror plane—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,

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

This relation can be immediately generalized to m-dimensional inner product spaces. Let the space V_m allow an orthogonal direct sum decomposition into a 1-dimensional and a (m−1)-dimensional subspace,

$V_{m}=V_{1}\oplus V_{m-1}$

and let v be an element of the one-dimensional space V₁ then the involution

$r\mapsto r-2v{\frac {(v,r)}{(v,v)}}$

is a reflection of r in the hyperplane V_m−1. (By definition a hyperplane is an m−1-dimensional linear subspace of a linear space of dimension m.) The inner product of two vectors v and w is notated as (v, w), which is common for vector spaces of arbitrary dimension.

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Fig. 2. The vector ${\vec {\mathbf {s} }}$ goes to ${\vec {\mathbf {s} }}\,'$ under reflection

Reflection in plane not through 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 normal to the plane. Obviously ${\vec {\mathbf {t} }}$ and ${\hat {\mathbf {n} }}$ are proportional, differ by a scaling.

Two consecutive reflections

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Fig. 3. Two reflections. Left drawing: 3-dimensional drawing. Right drawing: view along the PQ axis, drawing projected on the plane through ABC. This plane intersect the line PQ in the point P′

Two consecutive reflections in two intersecting planes give a rotation around the line of intersection. This is shown in Figure 2, 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.

From the point of view of matrices this result follows easily also. A reflection is represented by an improper matrix, that is, by an orthogonal matrix with determinant −1. The product of two orthogonal matrices is again an orthogonal matrix and the rule for determinants is det(AB) = det(A)det(B), so that the product of two improper rotation matrices is an orthogonal matrix with unit determinant, i.e., the matrix of a proper rotation.