# Reflection (geometry)

In Euclidean geometry, a **reflection** is a linear operation σ on ℝ^{3} with the property σ^{2} = E, the identity map. This property of σ is called *involution*. An involution is non-singular and is equal to its inverse: σ^{−1} = σ. Reflecting twice an arbitrary vector brings back the original vector:

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

It follows that reflection is symmetric: σ^{T} = σ. From the properties of determinants

follows that isometries have det(σ) = ±1. Those with determinant +1 are rotations; those with determinant −1 are reflections.

A reflection σ on ℝ^{3} has two sets of eigenvalues: {1, 1, −1} and {−1, −1, −1}. This follows because the eigenvalues of σ^{2} = E are +1 and hence the eigenvalues of σ are ±1. The product of the eigenvalues being the 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.

## [edit] Reflection in a plane

If is a unit vector normal (perpendicular) to a plane—the mirror plane—then is the projection of on this unit vector. From the figure it is evident that

If a non-unit normal is used then substitution of

gives the mirror image,

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

where **E** is the 3×3 identity matrix.

Dyadic products satisfy the matrix multiplication rule

By the use of this rule it is easily shown that

which confirms that reflection is involutory.

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

Use of the equation derived earlier gives

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

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

## [edit] 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 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 and of the second , then the rotation is represented by the matrix

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

This formula is used in vector rotation.