G G T = G T G = I 2.), with determinant 1 (the other possibility for orthogonal matrices is -1, which gives a mirror image, see below). each is a square matrix G whose transpose is its inverse, i.e. These matrices are the orthogonal matrices (i.e. First, a rotation around the origin is given by In terms of coordinates, rotations are most easily expressed by breaking them up into two operations. Rotations, denoted by R c,θ, where c is a point in the plane (the centre of rotation), and θ is the angle of rotation.This has the effect of shifting the plane in the direction of v. Translations, denoted by T v, where v is a vector in R 2.( Note: the notations for the types of isometries listed below are not completely standardised.) It can be shown that there are four types of Euclidean plane isometries (five if we include the identity). Where d( p, q) is the usual Euclidean distance between p and q.Ĭlassification of Euclidean plane isometries Such that for any points p and q in the plane, Neither are less drastic alterations like bending, stretching, or twisting.Īn isometry of the Euclidean plane is a distance-preserving transformation of the plane. However, folding, cutting, or melting the sheet are not considered isometries. There is one further type of isometry, called a glide reflection (see below under classification of Euclidean plane isometries). These are examples of translations, rotations, and reflections respectively. ![]() Notice that if a picture is drawn on one side of the sheet, then after turning the sheet upside down, we see the mirror image of the picture. Rotating the sheet by ten degrees around some marked point (which remains motionless). ![]()
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