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
- Refa(v) = − v, if v is parallel to a, and
- Refa(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 δij is the Kronecker delta.
The formula for the reflection in the affine hyperplane not through the origin is
Read more about this topic: Reflection (mathematics)
Famous quotes containing the words reflection and/or dimensions:
“It seems certain, that though a man, in a flush of humour, after intense reflection on the many contradictions and imperfections of human reason, may entirely renounce all belief and opinion, it is impossible for him to persevere in this total scepticism, or make it appear in his conduct for a few hours.”
—David Hume (1711–1776)
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)