AFFINE GEOMETRY. An affine geometry, as distin \ guished from a projective geometry, involves a definition of parallelism. Thus, when from the definitions and theorems of the ordinary euclidean geometry of the plane all those _involving euclidean measurement of length and angle are eliminated, the residue is an affine geometry. In this case the line at infinity is unique, in that any two lines meeting in a point of the former are parallel. In the projective geometry of the plane there is no such preferred line; when any line is designated as preferred in this sense, we have a corresponding affine geometry. For ex ample, in affine geometry there is a distinction between an ellipse, a parabola and a hyperbola, but not in projective geometry; moreover, only when the geometry is metrical is there a distinc tion between circles and ellipses.
The euclidean affine geometry is given analytic form by means of coordinates x and y referred to two intersecting lines of the plane in the usual manner. In this case the equations where the a's are arbitrary constants subject to the condition a12a21 0, define the most general transformation of points of the plane among themselves which transforms parallel lines into parallel lines. Such a transformation is called affine. The transformations (I) form a group, and we may say, following Klein, that the body of definitions and theorems expressing properties of quantities invariant under this group is an affine geometry. If x', y'; x", y"; , y"' are the coordinates of three non-collinear points, the quantity is invariant under transformations (I) for which a11a22—a12a21 = 1. These transformations are called equiaffine. There is a consider able body of theorems concerning properties of plane curves which remain invariant under equiaffine transformations. In particular, there is a metric theory based upon a certain invariant of the type (2) associated with a curve.
The foregoing concepts are readily generalized to spaces of higher order. Thus there is for spaces of three dimensions a theory of curves and surfaces analogous in many respects to the theory for euclidean space of three dimensions. The develop ments of these ideas have been made by many geometers, notably Klein, Mobius, Sylvester and lately by Blaschke and Pick.
Any n independent variables xi, where i takes the values to n, may be thought of as the coordinates of a general n dimensional space, or variety, in the sense that each set of values of the x's defines a point. In a space thus defined there is not a basis for the comparison of directions at different points. Such a basis is provided when a definition of parallelism is added. In Riemannian geometry we say, following Levi-Civita, that at points of a curve, defined by xi=fi(t), (i=i, • , n), a set of functions ti of t satisfying the system of equations are the components of a family of vectors parallel with respect to the curve; in this case Ftik are functions of the x's determined by the metric of the space (see RIEMANNIAN GEOMETRY). As thus defined, parallelism is relative to a curve, and not absolute, as in euclidean space. If the F's in (3) are taken as arbitrary func tions of the x's, equations (3) may be taken as the definition of parallelism for a general thus generalizing Levi-Civita's con cept of parallelism for spaces with a Riemannian metric. In this case we say that the space is affinely connected, following Weyl; the F's are called the coefficients of the affine connection.
When Fik the connection is called symmetric, otherwise asymmetric. When, and only when, the connection is symmetric, there exist preferred coordinate systems, with a given point as origin, with respect to which corresponding components of a vector at the given point and of a parallel one at a nearby point are the same. When the connection is asymmetric, there are other connections giving an equivalent definition of parallelism; the definition is unique for symmetric connections.
The fundamental curves of an affinely connected space are those whose tangents are parallel with respect to the curves; i.e., they are the straight lines of the space. They are the integral curves of the differential equations obtained when = dxi/dt in (3). Some writers call them the paths of the space, others geodesics, since they are the generalizations of the geodesics of Riemannian spaces. There are many affine connections with the same paths, so that the theorems dealing with the paths and not with a particular affine connection constitute a projective geometry. A theory of affinely connected spaces, in many respects analogous to Riemannian geometry, has been developed recently, notably by Cartan, Eddington, Einstein, Schouten, Joseph M. Thomas, Tracy Y. Thomas, Veblen and Weyl.
BIBLIOGRAPHY.-0. Veblen, Projective Geometry, vol. H. (Boston., Bibliography.-0. Veblen, Projective Geometry, vol. H. (Boston., 1918) ; H. Weyl, Space, Time, Matter (1922) ; W. Blaschke, Vorles sungen fiber Differential Geometric, vol. ii. (1923) J. A. Schouten, Der Ricci-Kalkiil (1924) ; L. P. Eisenhart, "Non-Riemannian Geom etry," American Mathematical Society (1927) ; 0. Veblen, "Invariants of Quadratic Differential Forms," Cambridge Tracts in Mathematics (1927). (L. P. E.)