Reflection (geometry)

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This editable Main Article is under development and subject to a disclaimer.

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In Euclidean geometry, a reflection is a linear operation σ on ℝ3 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 :

σ(𝐫→)=𝐫→′andσ(𝐫→′)=𝐫→.

The operation σ is an isometry of ℝ3 onto itself, which means that it preserves inner products and that its inverse is equal to its adjoint,

σT=σ−1(=σ).

Hence reflection is also symmetric: σT = 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.

PD Image
Fig. 1. The vector 𝐫→ goes to 𝐫→′ under reflection in a plane. The unit vector 𝐧^ is normal to mirror plane.

## Reflection in a plane[edit]

If 𝐧^ is a unit vector normal (perpendicular) to a plane--the mirror plane--then (𝐧^⋅𝐫→)𝐧^ is the projection of 𝐫→ on this unit vector. From the figure it is evident that

𝐫→−𝐫→′=2(𝐧^⋅𝐫→)𝐧^⟹𝐫→′=𝐫→−2(𝐧^⋅𝐫→)𝐧^

If a non-unit normal 𝐧→ is used then substitution of

𝐧^=𝐧→|𝐧→|≡𝐧→n

gives the mirror image,

𝐫→′=𝐫→−2(𝐧→⋅𝐫→)𝐧→n2

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

𝐫→′=[𝐄−2n2𝐧→⊗𝐧→]𝐫→,

where E is the 3×3 identity matrix.

Dyadic products satisfy the matrix multiplication rule

[𝐚→⊗𝐛→][𝐜→⊗𝐝→]=(𝐛→⋅𝐜→)(𝐚→⊗𝐝→).

By the use of this rule it is easily shown that

[𝐄−2n2𝐧→⊗𝐧→]2=𝐄,

which confirms that reflection is involutory.

PD Image
Fig. 2. The vector 𝐬→ goes to 𝐬→′ under reflection

## 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 𝐭→. The length of this vector is the distance from O to the plane. From Figure 2, we find

𝐫→=𝐬→−𝐭→,𝐫→′=𝐬→′−𝐭→

Use of the equation derived earlier gives

𝐬→′−𝐭→=𝐬→−𝐭→−2(𝐧^⋅(𝐬→−𝐭→))𝐧^.

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

𝐬→′=𝐬→−2(𝐧^⋅(𝐬→−𝐭→))𝐧^,

where 𝐧^ is a unit vector normal to the plane. Obviously 𝐭→ and 𝐧^ are proportional, they differ only by scaling. Therefore, the equation can be written solely in terms of 𝐭→,

𝐬→′=𝐬→−2𝐭→⋅(𝐬→−𝐭→)t2𝐭→,t2≡𝐭→⋅𝐭→.

## Two consecutive reflections[edit]

PD Image
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 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 𝐬→ and of the second 𝐭→, then the rotation is represented by the matrix

[𝐄−2t2𝐭→⊗𝐭→][𝐄−2s2𝐬→⊗𝐬→]=𝐄−2t2𝐭→⊗𝐭→−2s2𝐬→⊗𝐬→+4t2s2(𝐭→⋅𝐬→)(𝐭→⊗𝐬→)

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

δij−2titjt2−2sisjs2+4tisj(∑ktksk)t2s2.

This formula is used in vector rotation.