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A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.
To obtain exactly the same rotation (i.e. the same final coordinates of point P), the equivalent row vector must be post-multiplied by the transpose of R (i.e. wR T). Right- or left-handed coordinates The matrix and the vector can be represented with respect to a right-handed or left-handed coordinate system. Throughout the article, we assumed ...
The reflection hyperplane can be defined by its normal vector, a unit vector (a vector with length ) that is orthogonal to the hyperplane. The reflection of a point about this hyperplane is the linear transformation:
This is a reflection in the hyperplane perpendicular to v (negating any vector component parallel to v). If v is a unit vector, then Q = I − 2vv T suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size n × n can be constructed as a product of at most n such ...
Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions. In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis ) of an orthogonal transformation is an orthogonal matrix .
These equations can be proved through straightforward matrix multiplication and application of trigonometric identities, specifically the sum and difference identities.. The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group.
Point Q is the reflection of point P through the line AB. In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
The overall reflection of a layer structure is the sum of an infinite number of reflections. The transfer-matrix method is based on the fact that, according to Maxwell's equations , there are simple continuity conditions for the electric field across boundaries from one medium to the next.