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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.
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 ...
When the particle is in either of these configurations, it is no longer fixed under reflection in the y-axis: reflection swaps the two vacuum states. An example with continuous symmetry is given by a 3d analogue of the previous example, from rotating the graph around an axis through the top of the hill, or equivalently given by the graph f ( x ...
In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a 1 ⁄ 3 turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations.
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 ...
Thus reflection is a reversal of the coordinate axis perpendicular to the mirror's surface. Although a plane mirror reverses an object only in the direction normal to the mirror surface, this turns the entire three-dimensional image seen in the mirror inside-out, so there is a perception of a left-right reversal.
A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach. Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a semi-direct product of Z and C 2.
Geometrically speaking, the graph face of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis. Examples of even functions include |x|, x 2, x 4, cos(x), and cosh(x).