![]() ![]() Every point moves the same amount, twice the distance between the mirrors, and in the same direction. Two distinct mirrors that do not intersect must be parallel. Any two mirrors with the same fixed point and same angle give the same rotation, so long as they are used in the correct order. All other points rotate around it by twice the angle between the mirrors. Two distinct intersecting mirrors have a single point in common, which remains fixed. (In formal terms, topological orientation is reversed.) Points on the mirror are left fixed. Any pair of identical mirrors has the same effect.Īs Alice found through the looking-glass, a single mirror causes left and right hands to switch. Two reflections in the same mirror restore each point to its original position. In the Euclidean plane, we have the following possibilities. Thus isometries are an example of a reflection group. Reflections, or mirror isometries, can be combined to produce any isometry. A pure translation and a pure reflection are special cases with only two degrees of freedom, while the identity is even more special, with no degrees of freedom. This applies regardless of the details of the probability distribution, as long as θ and the direction of the added vector are independent and uniformly distributed and the length of the added vector has a continuous distribution. In all cases we multiply the position vector by an orthogonal matrix and add a vector if the determinant is 1 we have a rotation, a translation, or the identity, and if it is −1 we have a glide reflection or a reflection.Ī "random" isometry, like taking a sheet of paper from a table and randomly laying it back, " almost surely" is a rotation or a glide reflection (they have three degrees of freedom). It is the only isometry which belongs to more than one of the types described above. The identity isometry, defined by I( p) = p for all points p is a special case of a translation, and also a special case of a rotation. This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the combination is itself just a reflection in a parallel line. That is, we obtain the same result if we do the translation and the reflection in the opposite order.)Īlternatively we multiply by an orthogonal matrix with determinant −1 (corresponding to a reflection in a line through the origin), followed by a translation. 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 over, we see the mirror image of the picture. Turning the sheet over to look at it from behind.Rotating the sheet by ten degrees around some marked point (which remains motionless).Shifting the sheet one inch to the right. ![]() For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk. Informally, a Euclidean plane isometry is any way of transforming the plane without "deforming" it. It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections. The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. There are four types: translations, rotations, reflections, and glide reflections (see below under Classification § Notes). In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length.
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