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 classification of Euclidean plane isometries). Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Thales (circa 624-547 BC) dealing with spatial relationships. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant under the transformation. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ...

The set of Euclidean plane isometries forms a group under composition, which is the two-dimensional case of the Euclidean group. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...

Informal discussion

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 upside down. Notice that if a picture is drawn on one side of the sheet, then after turning the sheet upside down, the picture will appear to be "backwards". This is similar to the effect of viewing an object in a mirror.

These are examples of translations, rotations, and reflections respectively. There is one further type of isometry, called a glide reflection (see below under classifcation of Euclidean plane isometries). In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... In linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant under the transformation. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ...

However, folding, cutting, or burning the paper are not considered isometries.

Formal definition

An isometry (or rigid motion) of the Euclidean plane is a distance-preserving transformation of the plane. That is, it is a map In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

such that for any points p and q in the plane,

where d(p, q) is the usual Euclidean distance between p and q. The Euclidean distance of two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is computed as It is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...

Classification of Euclidean plane isometries

It can be shown that there are four types of Euclidean plane isometries. (Warning: the notations for the four types of isometries listed below are not completely standardised.)

• 

  Translation
  Translations, denoted by T_v, where v is a vector in R². This has the effect of shifting the plane in the direction of v. That is, for any point p in the plane,

or in terms of (x, y) coordinates,

• 

  Rotation
  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. In terms of coordinates, rotations are most easily expressed by breaking them up into two operations. First, a rotation around the origin (see coordinate rotation) is given by

A rotation around c can be accomplished by first translating c to the origin, then performing the rotation around the origin, and finally translating the origin back to c. That is,

or in other words,

• 

  Reflection
  Reflections, denoted by F_c,v, where c is a point in the plane and v is a unit vector in R². (F is for "flip".) This has the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c. To find a formula for F_c,v, we first use the dot product to find the component t of p − c in the v direction,

and then we obtain the reflection of p by subtraction,

• 

  Glide reflection
  Glide reflections, denoted by G_c,v,w, where c is a point in the plane, v is a unit vector in R², and w is a vector perpendicular to v. This is a combination of a reflection in the line described by c and v, followed by a translation along w. That is,

or in other words,

(It is also true that

that is, we obtain the same result if we do the translation and the reflection in the opposite order.)

Note that the identity isometry, defined by I(p) = p for all points p, is strictly speaking both a translation and a rotation. Also, every reflection is strictly speaking a glide reflection. Apart from these exceptions, the four categories above are mutually exclusive. In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ... In linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant under the transformation. ... In linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant under the transformation. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ... In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a sesquilinear function (·) : V × V → F, where V is a vector space over the field F, having some further properties. ... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ... In mathematics, an identity is an equality that remains true regardless of the values of any variables that appear within it. ...

Isometries as reflections

Reflections, or mirror isometries, can be combined to produce any isometry. In the plane, we have the following possibilities.

[d  ] Identity

Two reflections in the same mirror restore each point to its original position. Any pair of identical mirrors has the same effect.

[db] Reflection

As Alice found in Wonderland, left and right hands switch. (In formal terms, topological orientation is reversed.)

[dp] Rotation

Two distinct intersecting mirrors have a single point in common, which remains fixed. All other points rotate around it by twice the angle between the mirrors. 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.

[dd] Translation

Two distinct mirrors that do not intersect must be parallel. Every point moves the same amount, twice the distance between the mirrors. Any two mirrors with the same parallel direction and the same distance apart give the same translation, so long as they are used in the correct order.

[dq] Glide reflection

Three mirrors also send Alice to Wonderland. If they are all parallel, the effect is the same as a single mirror (slide a pair to cancel the third). Otherwise we can find an equivalent arrangement where two are parallel and the third is perpendicular to them. The effect is a reflection combined with a translation parallel to the mirror.

Adding more mirrors does not add more possibilities (in the plane), because they can always be rearranged to cause cancellation. John Tenniels illustration for A Mad Tea-Party, 1865 Alices Adventures in Wonderland is a work of childrens literature by the British mathematician and author Reverend Charles Lutwidge Dodgson under the pseudonym Lewis Carroll. ...

Isometries requiring an odd number of mirrors — reflection and glide reflection — always reverse left and right. The even isometries — identity, rotation, and translation — never do; they correspond to rigid motions, and form a normal subgroup of the full Euclidean group of isometries. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g-1ng is still in N. The statement N is a normal subgroup of G is written: . Another way... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...