counterclockwise for the other) are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles (otherwise, for example, all rhombi would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all rectangles would be similar). A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional. For given n, all regular n-gons are similar. Similar curves Several types of curves have the property that all examples of that type are similar to each other. These include: Lines (any two lines are even congruent) Line segments Circles Parabolas Hyperbolas of a specific eccentricity Ellipses of a specific eccentricity Catenaries Graphs of the logarithm function for different bases Graphs of the exponential function for different bases Logarithmic spirals are self-similar In Euclidean space A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection from the space onto itself that multiplies all distances by the same positive real number , so that for any two points and we have where "" is the Euclidean distance from to . The scalar has many names in the literature including; the ratio of similarity, the stretching factor and the similarity coefficient. When = 1 a similarity is called an isometry (rigid transformation). Two sets are called similar if one is the image of the other under a similarity. As a map , a similarity of ratio takes the form where is an orthogonal matrix and is a translation vector. Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments. Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve orientation and opposite similitudes change it. The similarities of Euclidean space form a group under the operation of composition called the similarities group . The direct similitudes form a normal subgroup of and the Euclidean group of isometries also forms a normal subgroup. The similarities group is itself a subgroup of the affine group, so every similarity is an affine transformation. One can view the Euclidean plane as the complex plane, that is, as a 2-dimensional space over the reals. The 2D similarity transformations can then be expressed in terms of complex arithmetic and are given by (direct similitudes) and (opposite similitudes), where and are