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The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal). The simplest nontrivial axial groups are equivalent to the abstract group Z 2: C i (equivalent to S 2) – inversion symmetry
Equivalently it is the largest fraction of the area of the shape that can be covered by a mirror reflection of the shape (with any orientation). A shape that is itself axially symmetric, such as an isosceles triangle , will have an axiality of exactly one, whereas an asymmetric shape, such as a scalene triangle , will have axiality less than one.
An axial dilation is depicted in Figure 1, in which two circles of opposite orientations undergo the same axial dilation. On lines, an axial dilation by t {\displaystyle t} units maps any line z {\displaystyle z} to a line z ′ {\displaystyle z'} such that z {\displaystyle z} and z ′ {\displaystyle z'} are parallel, and the perpendicular ...
Two distinct planes are either parallel or they intersect in a line. A line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the same plane must be parallel to each other. Two distinct planes perpendicular to the same line must be parallel to each other.
Desargues' theorem states that a central couple of triangles is axial. The converse statement, that an axial couple of triangles is central, is equivalent (either can be used to prove the other). Desargues' theorem can be proved in the real projective plane, and with suitable modifications for special cases, in the Euclidean plane.
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel.
Another type of non-Euclidean geometry is the hyperbolic plane, and arrangements of lines in this geometry have also been studied. [50] Any finite set of lines in the Euclidean plane has a combinatorially equivalent arrangement in the hyperbolic plane (e.g. by enclosing the vertices of the arrangement by a large circle and interpreting the ...