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Concept
Euclidean plane isometry
Résumé
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. There are four types: translations, rotations, reflections, and glide reflections (see below under ).
The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions. It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections.
Informally, a Euclidean plane isometry is any way of transforming the plane without "deforming" it. For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk. Examples of isometries include:
Shifting the sheet one inch to the right.
Rotating the sheet by ten degrees around some marked point (which remains motionless).
Turning the sheet over to look at it from behind. Notice that if a picture is drawn on one side of the sheet, then after turning the sheet over, we see the of the picture.
These are examples of translations, rotations, and reflections respectively. There is one further type of isometry, called a glide reflection (see below under classification of Euclidean plane isometries).
However, folding, cutting, or melting the sheet are not considered isometries. Neither are less drastic alterations like bending, stretching, or twisting.
An isometry of the Euclidean plane is a distance-preserving transformation of the plane. That is, it is a map
such that for any points p and q in the plane,
where d(p, q) is the usual Euclidean distance between p and q.
It can be shown that there are four types of Euclidean plane isometries. (Note: the notations for the types of isometries listed below are not completely standardised.)
Reflections, or mirror isometries, denoted by Fc,v, where c is a point in the plane and v is a unit vector in R2. (F is for "flip".) have the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c.
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