Motion (geometry)

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. More generally, the term motion is a synonym for surjective isometry in metric geometry, including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners. Motions can be divided into direct and indirect motions. Direct, proper or rigid motions are motions like translations and rotations that preserve the orientation of a chiral shape. Indirect, or improper motions are motions like reflections, glide reflections and Improper rotations that invert the orientation of a chiral shape. Some geometers define motion in such a way that only direct motions are motions. In differential geometry, a diffeomorphism is called a motion if it induces an isometry between the tangent space at a manifold point and the tangent space at the image of that point. Isometry group Given a geometry, the set of motions forms a group under composition of mappings. This group of motions is noted for its properties. For example, the Euclidean group is noted for the normal subgroup of translations. In the plane, a direct Euclidean motion is either a translation or a rotation, while in space every direct Euclidean motion may be expressed as a screw displacement according to Chasles' theorem. When the underlying space is a Riemannian manifold, the group of motions is a Lie group. Furthermore, the manifold has constant curvature if and only if, for every pair of points and every isometry, there is a motion taking one point to the other for which the motion induces the isometry. The idea of a group of motions for special relativity has been advanced as Lorentzian motions. For example, fundamental ideas were laid out for a plane characterized by the quadratic form in American Mathematical Monthly.