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Typical examples of affine planes are Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance.In other words, an affine plane over the reals is a Euclidean plane in which one has "forgotten" the metric (that is, one does not talk of lengths nor of angle measures).
A projective plane with a translation line is called a translation plane and the affine plane obtained by removing the translation line is called an affine translation plane. While in general it is often easier to work with projective planes, in this context the affine planes are preferred and several authors simply use the term translation ...
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]
For example, if the affine transformation acts on the plane and if the determinant of is 1 or −1 then the transformation is an equiareal mapping. Such transformations form a subgroup called the equi-affine group. [13] A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance
Affine space is characterized by a notion of pairs of parallel lines that lie within the same plane but never meet each-other (non-parallel lines within the same plane intersect in a point). Given any line, a line parallel to it can be drawn through any point in the space, and the equivalence class of parallel lines are said to share a direction .
The affine plane of order three is a (9 4, 12 3) configuration. When embedded in some ambient space it is called the Hesse configuration. It is not realizable in the Euclidean plane but is realizable in the complex projective plane as the nine inflection points of an elliptic curve with the 12 lines incident with triples of these.
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 function : is called affine if it preserves affine combinations. So (+ +) = + + ()for any affine combination + + in A. The space of affine functions A* is a linear space. The dual vector space of A* is naturally isomorphic to an (n+1)-dimensional vector space F(A) which is the free vector space on A modulo the relation that affine combination in A agrees with affine combination in F(A).