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Point Q is the reflection of point P through the line AB. In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
In geometry, a glide reflection or transflection is a geometric transformation that consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation.
Reflection. Reflections, or mirror isometries, denoted by F c,v, where c is a point in the plane and v is a unit vector in R 2.(F is for "flip".) have the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c.
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
Definition: [7] The midpoint of two elements x and y in a vector space is the vector 1 / 2 (x + y). Theorem [ 7 ] [ 8 ] — Let A : X → Y be a surjective isometry between normed spaces that maps 0 to 0 ( Stefan Banach called such maps rotations ) where note that A is not assumed to be a linear isometry.
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 ...
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
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...