vector space then the augmented matrix that achieves this affine transformation is This formulation works irrespective of whether any of the domain, codomain and image vector spaces have the same number of dimensions. For example, the affine transformation of a vector plane is uniquely determined from the knowledge of where the three vertices () of a non-degenerate triangle are mapped to (), regardless of the number of dimensions of the codomain and regardless of whether the triangle is non-degenerate in the codomain. Properties Properties preserved An affine transformation preserves: collinearity between points: three or more points which lie on the same line (called collinear points) continue to be collinear after the transformation. parallelism: two or more lines which are parallel, continue to be parallel after the transformation. convexity of sets: a convex set continues to be convex after the transformation. Moreover, the extreme points of the original set are mapped to the extreme points of the transformed set. ratios of lengths of parallel line segments: for distinct parallel segments defined by points and , and , the ratio of and is the same as that of and . barycenters of weighted collections of points. Groups As an affine transformation is invertible, the square matrix appearing in its matrix representation is invertible. The matrix representation of the inverse transformation is thus The invertible affine transformations (of an affine space onto itself) form the affine group, which has the general linear group of degree as subgroup and is itself a subgroup of the general linear group of degree . The similarity transformations form the subgroup where is a scalar times an orthogonal matrix. 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. A transformation that is both equi-affine and a similarity is an isometry of the plane taken with Euclidean distance. Each of these groups has a subgroup of orientation-preserving or positive affine transformations: those where the determinant of is positive. In the last case this is in 3D the group of rigid transformations (proper rotations and pure translations). If there is a fixed point, we can take that as the origin, and the affine transformation reduces to a linear transformation. This may make it easier to classify and understand the transformation. For example, describing a