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A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in ...
An xy-Cartesian coordinate system rotated through an angle to an x′y′-Cartesian coordinate system In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and ...
, = [ ] []. which is a reflection in the x-axis followed by a rotation by an angle θ, or equivalently, a reflection in a line making an angle of θ/2 with the x-axis. Reflection in a parallel line corresponds to adding a vector perpendicular to it.
Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where
The dihedral group D 2 is generated by the rotation r of 180 degrees, and the reflection s across the x-axis. The elements of D 2 can then be represented as {e, r, s, rs}, where e is the identity or null transformation and rs is the reflection across the y-axis. The four elements of D 2 (x-axis is vertical here) D 2 is isomorphic to the Klein ...
This isometry maps the x-axis to itself; any other line which is parallel to the x-axis gets reflected in the x-axis, so this system of parallel lines is left invariant. The isometry group generated by just a glide reflection is an infinite cyclic group. [1]
A spatial rotation is a linear map in one-to-one correspondence with a 3 × 3 rotation matrix R that transforms a coordinate vector x into X, that is Rx = X. Therefore, another version of Euler's theorem is that for every rotation R , there is a nonzero vector n for which Rn = n ; this is exactly the claim that n is an eigenvector of R ...
In geometry, an improper rotation [1] (also called rotation-reflection, [2] rotoreflection, [1] rotary reflection, [3] or rotoinversion [4]) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each a