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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).
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 ...
In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S.Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space.
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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 ...
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In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V.The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can ...
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]