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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 ...
A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L 1. Then reflect P′ to its image P′′ on the other side of line L 2. If lines L 1 and L 2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the ...
To give an example of the usefulness of this, suppose we wish to find a plane orthogonal to a certain line L in 3D and passing through a certain point P. L is a 2-reflection and is a 3-reflection, so taking their geometric product PL in some sense produces a 5-reflection; however, as in the picture below, two of these reflections cancel ...
Call the images of p 2 and p 3 under this reflection p 2 ′ and p 3 ′. If q 2 is distinct from p 2 ′, bisect the angle at q 1 with a new mirror. With p 1 and p 2 now in place, p 3 is at p 3 ″; and if it is not in place, a final mirror through q 1 and q 2 will flip it to q 3. Thus at most three reflections suffice to reproduce any plane ...
Two different reflections in two dimensions generating a rotation. Every simple rotation can be generated by two reflections. Reflections can be specified in n dimensions by giving an (n − 1)-dimensional subspace to reflect in, so a two-dimensional reflection is in a line, a three-dimensional reflection is in a plane, and so on. But this ...
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 ...
Any transformation of the plane leaving this pattern invariant can be decomposed into a translation, (x, y) ↦ (n + x, y), optionally followed by a reflection in either the horizontal axis, (x, y) ↦ (x, −y), or the vertical axis, (x, y) ↦ (−x, y), provided that this axis is chosen through or midway between two dots, or a rotation by ...
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.