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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 term reflection is loose, and considered by some an abuse of language, with inversion preferred; however, point reflection is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identity map – which is also true of other maps called reflections.
A point with a reflection coefficient magnitude 0.63 and angle 60° represented in polar form as , is shown as point P 1 on the Smith chart. To plot this, one may use the circumferential (reflection coefficient) angle scale to find the ∠ 60 ∘ {\displaystyle \angle 60^{\circ }\,} graduation and a ruler to draw a line passing through this and ...
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
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 ...
A reflection in a line is an opposite isometry, like R 1 or R 2 on the image. Translation T is a direct isometry: a rigid motion. [1] In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
If x is a reflection point (0, 5, 10, 15, 20, or 25), its stabilizer is the group of order two containing the identity and the reflection in x. In other cases the stabilizer is the trivial group. For a fixed x in X, consider the map from G to X given by g ↦ g · x. The image of this map is the orbit of x and the coimage is the set of all left ...
On one side, each point is contained in three light rays; on the other side, each point is contained in one light ray. In differential geometry, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the concept of caustics in geometric optics. The ray's source may be a point (called the radiant) or ...