The parallel shift or translation is a geometric mapping that shifts every point of the plane of the drawing or space in the same direction by the same distance. It can be identified by a vector , the so-called displacement vector .
Parallel displacements are part of the movements , since lengths and angles are retained when they are used. As movements, they - especially the parallel displacements in the plane - are also counted among the congruence maps.
The concept of parallel displacement can be generalized from the two- or three-dimensional visual space into the n-dimensional Euclidean space and even further into the Riemannian geometry or the affine geometry .
Two-dimensional viewing space
In two-dimensional (Euclidean) space , a parallel shift is a mathematical function that shifts every point in space by the same distance in the same direction. A parallel shift is thus given by an affine linear function
where applies, described.
In Riemannian geometry , the concept of parallel displacement from Euclidean geometry is generalized to curved objects such as the surface of a sphere. Mathematically precisely, these curved objects are defined as Riemannian manifolds . Vectors at these manifolds can be shifted in parallel along curves. This method was precisely formulated by Tullio Levi-Civita . Today it is mostly referred to as parallel transport but also as parallel displacement.
- Straight lines are mapped onto parallel straight lines.
- If any point is changed at all, the figure has no fixed point.
- T 1 :
- T 2 :
Here, too, translation is always an affinity in the sense of synthetic geometry. The continuation of a translation in the projective closure of the affine space is a projective perspective and therefore a projectivity .
When defining the term parallel translation or translation , different emphases are set in different areas of geometry and linear algebra, whereby the generalized definition is valid everywhere. Please refer
- on the concept of parallel displacement in the elementary geometry of the plane of the drawing : congruence mapping .
- on the concept of parallel displacement in three-dimensional visual space: movement (mathematics) .
- on the concept of translation in synthetic geometry : Affine translation level .
- on linear motion in physics: translation (physics)
Parallel shift in linear algebra and planar and spatial geometry:
- Uwe Storch, Hartmut Wiebe: Textbook of Mathematics, Volume II: Linear Algebra . BI-Wissenschafts-Verlag, 1990, ISBN 3-411-14101-8
Translation in synthetic geometry:
- Wendelin Degen and Lothar Profke: Fundamentals of affine and Euclidean geometry , Teubner, Stuttgart, 1976, ISBN 3-519-02751-8
- Günter Pickert : Level incidence geometry. 2nd edition, Frankfurt am Main 1968
- Sword (1976)