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A path isometry or arcwise isometry is a map which preserves the lengths of curves; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. [5] [6] This term is often abridged to simply isometry, so one should take care to determine from context which type is intended.
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations , rotations , reflections , and glide reflections (see below § Classification ).
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. [1] [self-published source] [2] [3] The rigid transformations include rotations, translations, reflections, or any sequence of ...
A Euclidean isometry can be direct or indirect, depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, often denoted SE(n) and E + (n), whose elements are called rigid motions or Euclidean motions. They comprise arbitrary combinations of translations and ...
In geometry, the Beckman–Quarles theorem states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. Equivalently, every homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane. The theorem is named ...
In particular, isometries and local isometries of surfaces preserve Gaussian curvature. [66] This theorem can expressed in terms of the power series expansion of the metric, ds, is given in normal coordinates (u, v) as ds 2 = du 2 + dv 2 − K(u dv – v du) 2 /12 + ….
An isometry between two metric spaces is a bijection preserving the distance, [b] that is ((), ()) = (,). In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm ‖ ‖ = ‖ ‖,
It preserves the curvature tensor at the tangent space of every point on (), that is, it preserves how the infinitesimal patches fit together. If f {\displaystyle f} is an isometry, it must preserve the geodesics.