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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. The group has an identity: Rot(0). Every rotation Rot(φ) has an inverse Rot(−φ). Every reflection Ref(θ) is its own inverse. Composition has closure and is ...
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, + + + =.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane. Some mathematicians use "flip" as a synonym for "reflection". [2 ...
Alternatively we multiply by an orthogonal matrix with determinant −1 (corresponding to a reflection in a line through the origin), followed by a translation. This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the combination is itself just a reflection in a ...
By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation.
Thus, the determinant of a rotation orthogonal matrix must be 1. The only other possibility for the determinant of an orthogonal matrix is −1, and this result means the transformation is a hyperplane reflection, a point reflection (for odd n), or another kind of improper rotation. Matrices of all proper rotations form the special orthogonal ...
In mathematics, reflection through the origin refers to the point reflection of Euclidean space R n across the origin of the Cartesian coordinate system. Reflection through the origin is an orthogonal transformation corresponding to scalar multiplication by − 1 {\displaystyle -1} , and can also be written as − I {\displaystyle -I} , where I ...