Reflection (mathematics) - Reflection Through A Hyperplane in n Dimensions

Reflection Through A Hyperplane in n Dimensions

Given a vector a in Euclidean space Rn, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by

where v·a denotes the dot product of v with a. Note that the second term in the above equation is just twice the vector projection of v onto a. One can easily check that

• Ref_a(v) = − v, if v is parallel to a, and
• Ref_a(v) = v, if v is perpendicular to a.

Using the geometric product the formula is a little simpler

Since these reflections are isometries of Euclidean space fixing the origin, they may be represented by orthogonal matrices. The orthogonal matrix corresponding to the above reflection is the matrix whose entries are

where δ_ij is the Kronecker delta.

The formula for the reflection in the affine hyperplane not through the origin is