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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 ...
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 ...
This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the combination is itself just a reflection in a parallel line. The identity isometry, defined by I(p) = p for all points p is a special case of a translation, and also a special case of a rotation. It is the ...
(A reflection would not preserve handedness; for instance, it would transform a left hand into a right hand.) To avoid ambiguity, a transformation that preserves handedness is known as a rigid motion, a Euclidean motion, or a proper rigid transformation. In dimension two, a rigid motion is either a translation or a rotation.
A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a translation. A motion of a Euclidean space is the same as its isometry : it leaves the distance between any two points unchanged after the transformation.
Any motion is a one-to-one mapping of space R onto itself such that every three points on a line will be transformed into (three) points on a line. The identical mapping of space R is a motion. The product of two motions is a motion. The inverse mapping of a motion is a motion.
rotation by 180° about an axis and reflection in a plane through that axis; rotation by 180° about an axis and rotation by 180° about a perpendicular axis (results in rotation by 180° about the axis perpendicular to both) two rotoreflections about the same axis, with respect to the same plane; two glide reflections with respect to the same ...
inversion through a point (half turn) — two reflections through mutually perpendicular lines passing through the given point, i.e. a rotation of 180 degrees around the point; two degrees of freedom. rotation around a normal point — two reflections through lines passing through the given point (includes inversion as a special case); points ...