In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. It is the geometry of affine space, of a given dimension n over a field K. The case where K is the real numbers gives an adequate idea of the content. Geometry (Greek Γεωμετρια, geo = earth, metria = measure (check accuracy of this)) arose as the field of knowledge dealing with spatial relationships. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ...

Intuitive background

Affine geometry can be explained as the geometry of vectors, not involving any notions of length or angle. Affine space is distinguished from a vector space of the same dimension by 'forgetting' the origin 0. That way of thinking was in older texts sometimes talked about as a theory of free vectors. A contemporary and more abstract way of putting it is mentioned at the end of this page, completing a formal reduction of affine geometry to linear algebra. In physics and engineering, the word vector typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but this... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...

Applications and relationships

The notions of affine geometry have applications, for example in differential geometry. Because of the close relation with linear algebra, they are not so often isolated. There do exist several ways, rather than just one, of expressing that relationship. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...

Affine transformations

Main article: affine transformation. In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation. ...

According to the general scheme of the Erlangen programme, we can say best what affine geometry is by looking at the underlying group of symmetry transformations. An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

This can be done quickly in terms of a vector space V. The general linear group GL(V) is not the whole affine group: we must allow also translations by vectors v in V. (Such a translation maps any w in V to w + v.) The affine group is generated by the general linear group and the translations and is in fact their semidirect product V  GL(V). (Here we think of V as a group under its operation of addition, and use the defining representation of GL(V) on V to define the semidirect product.) In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

Affine theorems

We therefore identify as affine theorems any geometric results that can be stated in terms invariant under the affine group. An example from the plane geometry of triangles is the theorem about the concurrence of the lines joining each vertex to the mid-point of the opposite side (at the centroid or barycentre). The idea of mid-point is an affine invariant. There are other classical examples (theorems of Ceva, Menelaus). Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ... The barycenter (from the Greek βαρύκεντρον) is the center of mass of two or more bodies which are orbiting each other, and is the point around which both of them orbit. ... Cevas Theorem (pronounced Cheva) is a very popular theorem in elementary geometry. ... Menelaus theorem (also known as Menelaus theorem, Menelauss theorem, as well as theorem of Menelaus; attributed to Menelaus of Alexandria) is a theorem about triangles in plane geometry. ...

These theorems are notable for having proofs by vector methods. Notice that the logic runs in one direction: if a theorem is an affine theorem, there is no reason why it shouldn't be proved by vectors. It doesn't yet run the other way, that there must be such a proof. That may be desirable from a geometric point of view, rather than finding a heavy-handed proof using analytic geometry. But it's then a question of axiomatic study (so-called synthetic point of view). Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems. ...

Affine invariants can also assist calculations. For example, the lines that divide the area of a triangle into two equal halves form an envelope inside the triangle. The ratio of the area of the envelope to the area of the triangle is affine invariant, and so only needs to be calculated from a simple case such as a unit isosceles right angled triangle to give , i.e. 0.019860... or less than 2%, for all triangles.

What is affine space?

The term affine space is used in projective geometry as the complement of the points (hyperplane) at infinity (see also projective space). There is an implied usage made above: affine space is the space discussed in affine geometry. And there is a third way of defining it, starting from a vector space. Actually the space of translations in an affine space gives back a copy of the underlying vector space anyway. What is required to give a consistent dictionary between all these ways of talking about affine space is the construction of the affine space of a vector space. In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In mathematics, a projective space is a fundamental construction from any vector space. ...

Observe that in combinations of vectors v − w the result is unchanged by translation (v moves as much in one direction as − w does in the other) Computationally one must simply restrict discussion to linear combinations of vectors with sum of coefficients equal to zero: these have the same property, and are exactly the sums that can be expressed as combinations of simple differences v − w. This tells us one way to explain the concept of affine space: it's a vector space with the subtraction and scalar multiplication operations. That is one precise way in which to 'forget the origin'.

The abstract definition

This is concise and ultimately more successful (at a price). For any group G there is a notion of principal homogeneous space for G: a set S on which G acts in a way isomorphic to the way it permutes itself by multiplication. An affine space A for a vector space V is just such a principal homogeneous space; one then has to recover scalar multiplication on A as a well-defined concept. In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ...