Euclidean plane isometry

Ask a question about 'Euclidean plane isometry' Start a new discussion about 'Euclidean plane isometry' Answer questions from other users Full Discussion Forum

Informal discussion

• 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, we see the mirror image
  Mirror Image
  
  "Mirror Image" is an episode of the television series The Twilight Zone ....
   of the picture.

Formal definition

Classification of Euclidean plane isometries

• 
  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,

• Rotations
  Rotation (mathematics)
  
  In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point....
  , denoted by R_c,?, where c is a point in the plane (the centre of rotation), and ? 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 is given by

These matrices are the orthogonal matrices

Orthogonal matrix

In matrix theory, a real number orthogonal matrix is a Matrix #Square matrices Q whose transpose is its inverse matrix:A special orthogonal matrix is an orthogonal matrix with determinant +1:...
 (i.e. each is a square matrix G whose transpose

Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
 is its inverse, i.e. ), with determinant 1 (the other possibility for orthogonal matrices is -1, which gives a mirror image, see below). They form the special orthogonal group

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
 SO(2).

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,

Alternatively, a rotation around the origin is performed, followed by a translation:

• 
  Reflection
  Reflection (mathematics)
  
  In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
  s, or mirror isometries, denoted by F_c,v, where c is a point in the plane and v is a unit vector
  Unit vector
  
  In mathematics, a unit vector in a normed vector space is a Vector space whose Norm is 1 . A unit vector is often denoted by a lowercase letter with a superscribed caret or ?hat?, like this: ....
   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. The line L is called the reflection axis or the associated mirror. To find a formula for F_c,v, we first use the dot product
  Dot product
  
  In mathematics, the dot product, also known as the scalar product, is an operation which takes two vector over the real numbers R and returns a real-valued scalar quantity....
   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 reflection
  
  In geometry, a glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line....
  s, 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,

Alternatively we multiply by an orthogonal matrix with determinant -1 (corresponding to a reflection in a line through the origin), followed by a translation. 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.

Isometries as reflection group

Mirror combinations

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

• ; [db] Reflection

As Alice found through the looking-glass

Through the Looking-Glass

Through the Looking-Glass, and What Alice Found There is a work of children's literature by Lewis Carroll , generally categorized as literary nonsense....
, a single mirror causes left and right hands to switch. (In formal terms, topological orientation is reversed.) Points on the mirror are left fixed. Each mirror has a unique effect.

• ; [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, and in the same direction. No points are left fixed. 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 entertain Alice (switch hands). 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. No points are left fixed.

Three mirrors suffice

Proof. An isometry is completely determined by its effect on three independent (not collinear) points. So suppose p₁, p₂, p₃ map to q₁, q₂, q₃; we can generate a sequence of mirrors to achieve this as follows. If p₁ and q₁ are distinct, choose their perpendicular bisector as mirror. Now p₁ maps to q₁; and we will pass all further mirrors through q₁, leaving it fixed. Call the images of p₂ and p₃ under this reflection p₂′ and p₃′. If q₂ is distinct from p₂′, bisect the angle at q₁ with a new mirror. With p₁ and p₂ now in place, p₃ is at p₃′′; and if it is not in place, a final mirror through q₁ and q₂ will flip it to q₃. Thus at most three reflections suffice to reproduce any plane isometry. ?

Recognition

>

		Preserves hands?
		Yes	No
Fixed point?	Yes	Rotation	Reflection
	No	Translation	Glide reflection

Group structure

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

Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
 of the full Euclidean group

Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
 of isometries. Neither the full group nor the even subgroup are abelian

Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
; for example, reversing the order of composition of two parallel mirrors reverses the direction of the translation they produce.

Proof. The identity is an isometry; nothing changes, so distance cannot change. And if one isometry cannot change distance, neither can two (or three, or more) in succession; thus the composition of two isometries is again an isometry, and the set of isometries is closed under composition. The identity isometry is also an identity for composition, and composition is associative; therefore isometries satisfy the axioms for a semigroup

Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a nonempty Set S together with an associative binary operation. In other words, a semigroup is an associative Magma ....
. For a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
, we must also have an inverse for every element. To cancel a reflection, we merely compose it with itself. (Reflections are involution

Involution

In mathematics, an involution, or an involutary function, is a function that is its own inverse function, so that...
s.) And since every isometry can be expressed as a sequence of reflections, its inverse can be expressed as that sequence reversed. Notice that the cancellation of a pair of identical reflections reduces the number of reflections by an even number, preserving the parity of the sequence; also notice that the identity has even parity. Therefore all isometries form a group, and even isometries a subgroup. (Odd isometries do not include the identity, so are not a subgroup.) This subgroup is a normal subgroup, because sandwiching an even isometry between two odd ones yields an even isometry. ?

Since the even subgroup is normal, it is the kernel

Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective....
 of a homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
 to a quotient group

Quotient group

In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element....
, where the quotient is isomorphic to a group consisting of a reflection and the identity. However the full group is not a direct product

Direct product

In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
, but only a semidirect product

Semidirect product

In mathematics, especially in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup....
, of the even subgroup and the quotient group.

Composition

Composition of isometries mixes kinds in assorted ways. We can think of the identity as either two mirrors or none; either way, it has no effect in composition. And two reflections give either a translation or a rotation, or the identity (which is both, in a trivial way). Reflection composed with either of these could cancel down to a single reflection; otherwise it gives the only available three-mirror isometry, a glide reflection. A pair of translations always reduces to a single translation; so the challenging cases involve rotations. We know a rotation composed with either a rotation or a translation must produce an even isometry. Composition with translation produces another rotation (by the same amount, with shifted fixed point), but composition with rotation can yield either translation or rotation. It is often said that composition of two rotations produces a rotation, and Euler proved a theorem to that effect in 3D; however, this is only true for rotations sharing a fixed point.

Translation, rotation, and orthogonal subgroups

We thus have two new kinds of isometry subgroups: all translations, and rotations sharing a fixed point. Both are subgroups of the even subgroup, within which translations are normal. Because translations are a normal subgroup, we can factor them out leaving the subgroup of isometries with a fixed point, the orthogonal group

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
.

:Proof. If two rotations share a fixed point, then we can swivel the mirror pair of the second rotation to cancel the inner mirrors of the sequence of four (two and two), leaving just the outer pair. Thus the composition of two rotations with a common fixed point produces a rotation by the sum of the angles about the same fixed point.

If two translations are parallel, we can slide the mirror pair of the second translation to cancel the inner mirror of the sequence of four, much as in the rotation case. Thus the composition of two parallel translations produces a translation by the sum of the distances in the same direction. Now suppose the translations are not parallel, and that the mirror sequence is A₁, A₂ (the first translation) followed by B₁, B₂ (the second). Then A₂ and B₁ must cross, say at c; and, reassociating, we are free to pivot this inner pair around c. If we pivot 90°, an interesting thing happens: now A₁ and A₂′ intersect at a 90° angle, say at p, and so do B₁′ and B₂, say at q. Again reassociating, we pivot the first pair around p to make B₂″ pass through q, and pivot the second pair around q to make A₁″ pass through p. The inner mirrors now coincide and cancel, and the outer mirrors are left parallel. Thus the composition of two non-parallel translations also produces a translation. Also, the three pivot points form a triangle whose edges give the head-to-tail rule of vector addition: 2(p c) + 2(c q) = 2(p q). ?

Nested group construction

The subgroup structure suggests another way to compose an arbitrary isometry:

Pick a fixed point, and a mirror through it.

1. If the isometry is odd, use the mirror; otherwise do not.
2. If necessary, rotate around the fixed point.
3. If necessary, translate.

This works because translations are a normal subgroup of the full group of isometries, with quotient the orthogonal group; and rotations about a fixed point are a normal subgroup of the orthogonal group, with quotient a single reflection.

Discrete subgroups

The subgroups discussed so far are not only infinite, they are also continuous (Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
s). Any subgroup containing at least one non-zero translation must be infinite, but subgroups of the orthogonal group can be finite. For example, the symmetries

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance; such that it reflects beauty or perfection....
 of a regular pentagon

Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
 consist of rotations by integer multiples of 72° (360° / 5), along with reflections in the five mirrors which perpendicularly bisect the edges. This is a group, D₅, with 10 elements. It has a subgroup, C₅, of half the size, omitting the reflections. These two groups are members of two families, D_n and C_n, for any n > 1. Together, these families constitute the rosette group

Point group

In mathematics, a point group is a group of geometric symmetry leaving a point fixed....
s.

Translations do not fold back on themselves, but we can take integer multiples of any finite translation, or sums of multiples of two such independent translations, as a subgroup. These generate the lattice

Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which linear span the real number vector space Rn....
 of a periodic tiling

Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
 of the plane.

We can also combine these two kinds of discrete groups — the discrete rotations and reflections around a fixed point and the discrete translations — to generate the frieze group

Frieze group

A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetry in the pattern....
s and wallpaper group

Wallpaper group

A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry in the pattern. Such patterns occur frequently in architecture and decorative art....
s. Curiously, only a few of the fixed-point groups are found to be compatible

Crystallographic restriction theorem

The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetry of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold....
 with discrete translations. In fact, lattice compatibility imposes such a severe restriction that, up to isomorphism

Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
, we have only 7 distinct frieze groups and 17 distinct wallpaper groups. For example, the pentagon symmetries, D₅, are incompatible with a discrete lattice of translations. (Each higher dimension also has only a finite number of such crystallographic groups, but the number grows rapidly; for example, 3D has 320 groups and 4D has 4783.)

Isometries in the complex plane

In terms of complex numbers, the isometries of the plane are addition of a complex constant (translation), multiplication by a complex constant with modulus 1 (rotation), complex conjugation (reflection in the real axis), and combinations.

See also

• Coordinate rotations and reflections
  Coordinate rotations and reflections
  
  In geometry, 2D coordinate rotations and reflection s are two kinds of Euclidean plane isometry which are related to one another.A rotation in the plane can be formed by composing a pair of reflections....
  
• Hjelmslev's theorem
  Hjelmslev's theorem
  
  In geometry, Hjelmslev's theorem, named after Johannes Hjelmslev, is the statement that if points P, Q, R... of one and the same line are isometrically mapped to points P?, Q?, R?......
  

External links