Fixed element

This article explains the concept of geometry, for which the combinatorics see fixed-point-free permutation ; for the concept of structural engineering, see connecting element .

In geometry, fixed elements of a mapping are quantities of the domain that are mapped onto themselves. They include:

• Fixed points ${\displaystyle P\mapsto P.}$
• Fixed point lines for all points of a line g . All points of the straight line are therefore fixed points in the figure. ${\displaystyle P\mapsto P}$
• Fixed straight lines (but not essentialfor, for example, reversing the orientation: there is only one fixed point; fixed point straight lines are special fixed lines) ${\displaystyle g\mapsto g}$ ${\displaystyle P\mapsto P}$ ${\displaystyle P\in g}$
• Fixed circle of inversion for , the unit circle - here too strict and less strict form exist, the example gives the strict form point by point for everyone ${\displaystyle k\mapsto k'={\tfrac {1}{k}}=k}$ ${\displaystyle k:r=1,M=[0,0]}$ ${\displaystyle P\in k.}$
• Fixed levels in spatial problems
• where the descriptive terms of geometry fail with more than three-dimensional problems, one usually speaks only of fixed elements
• Fixed point hyperplanes are important for the classification of the affinities and projectivities : Subspaces of the depicted spaces are called, the dimensions of which are one smaller than that of the entire space if they remain fixed point by point in an image.

Fixed elements are the axes of symmetry (or points and other elements) of a geometric symmetry .

literature

• Fixed element . In: H. Athens, J. Bruhn (Hrsg.): Lexikon der Schulmathematik . [License] study edition. 2 F-K . Weltbildverlag, Augsburg 1994, ISBN 3-89350-174-6 , p. 287 f .