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:
“With respect to a true culture and manhood, we are essentially provincial still, not metropolitan,—mere Jonathans. We are provincial, because we do not find at home our standards; because we do not worship truth, but the reflection of truth; because we are warped and narrowed by an exclusive devotion to trade and commerce and manufacturers and agriculture and the like, which are but means, and not the end.”
—Henry David Thoreau (1817–1862)
“The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
—Thomas Jefferson (1743–1826)