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:
“In the last analysis, love is only the reflection of a man’s own worthiness from other men.”
—Ralph Waldo Emerson (1803–1882)
“It seems to me that we do not know nearly enough about ourselves; that we do not often enough wonder if our lives, or some events and times in our lives, may not be analogues or metaphors or echoes of evolvements and happenings going on in other people?—or animals?—even forests or oceans or rocks?—in this world of ours or, even, in worlds or dimensions elsewhere.”
—Doris Lessing (b. 1919)