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The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group. The group has an identity: Rot(0). Every rotation Rot(φ) has an inverse Rot(−φ). Every reflection Ref(θ) is its own inverse. Composition has closure and is ...
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.
(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.
Earth's rotation axis moves with respect to the fixed stars (inertial space); the components of this motion are precession and nutation. It also moves with respect to Earth's crust; this is called polar motion. Precession is a rotation of Earth's rotation axis, caused primarily by external torques from the gravity of the Sun, Moon and other bodies.
The plane of rotation is the plane containing m and n, which must be distinct otherwise the reflections are the same and no rotation takes place. As either vector can be replaced by its negative the angle between them can always be acute, or at most π / 2 . The rotation is through twice the angle between the vectors, up to π or a
An exploration of transformation geometry often begins with a study of reflection symmetry as found in daily life. The first real transformation is reflection in a line or reflection against an axis. The composition of two reflections results in a rotation when the lines intersect, or a translation when they are parallel.
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.
Polar motion of the Earth is the motion of the Earth's rotational axis relative to its crust. [2]: 1 This is measured with respect to a reference frame in which the solid Earth is fixed (a so-called Earth-centered, Earth-fixed or ECEF reference frame). This variation is a few meters on the surface of the Earth.