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A continuous motion is a continuous path in (,) that describes the physical motion between two frameworks of (,) that preserves all constraints. A trivial motion is a continuous motion resulting from the ( d + 1 2 ) {\displaystyle d+1 \choose 2} Euclidean isometries , i.e. translations and rotations.
In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion. [1] More generally, the term motion is a synonym for surjective isometry in metric geometry, [2] including elliptic geometry and hyperbolic ...
(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.
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.
Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point and a translation. In one-dimensional space , there are only trivial rotations. In two dimensions , only a single angle is needed to specify a rotation about the origin – the angle of rotation that specifies an element of the circle group (also ...
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President-elect Donald Trump, after rejecting House Speaker Mike Johnson's plan to avoid a government shutdown, worked the phones on Thursday, showing wavering confidence in Johnson and claiming ...
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