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Plane_Crazy_(SILENT).webm (WebM audio/video file, VP9, length 6 min 0 s, 640 × 480 pixels, 1.9 Mbps overall, file size: 81.64 MB) This is a file from the Wikimedia Commons . Information from its description page there is shown below.
There are two ways to formally define affine planes, which are equivalent for affine planes over a field. The first way consists in defining an affine plane as a set on which a vector space of dimension two acts simply transitively. Intuitively, this means that an affine plane is a vector space of dimension two in which one has "forgotten ...
The usage of complex numbers (instead of the real numbers) does not lead to a Möbius plane, because in the complex affine plane the curve + = is not a circle-like curve, but a hyperbola-like one. Fortunately there are a lot of fields (numbers) together with suitable quadratic forms that lead to Möbius planes (see below).
A similar construction, starting from the projective plane of order 3, produces the affine plane of order 3 sometimes called the Hesse configuration. An affine plane of order n exists if and only if a projective plane of order n exists (however, the definition of order in these two cases is not the same). Thus, there is no affine plane of order ...
A plane is said to have the "minor affine Desargues property" when two triangles in parallel perspective, having two parallel sides, must also have the third sides parallel. If this property holds in the affine plane defined by a ternary ring, then there is an equivalence relation between "vectors" defined by pairs of points from the plane. [14]
A plane segment or planar region (or simply "plane", in lay use) is a planar surface region; it is analogous to a line segment. A bivector is an oriented plane segment, analogous to directed line segments. [a] A face is a plane segment bounding a solid object. [1] A slab is a region bounded by two parallel planes.
In the same way as in the real case, the plane may also be viewed as the simplest, one-dimensional (in terms of complex dimension, over the complex numbers) complex manifold, sometimes called the complex line. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold.
In mathematics, real projective space, denoted or (), is the topological space of lines passing through the origin 0 in the real space +. It is a compact , smooth manifold of dimension n , and is a special case G r ( 1 , R n + 1 ) {\displaystyle \mathbf {Gr} (1,\mathbb {R} ^{n+1})} of a Grassmannian space.