A geometrical symmetry is a geometrical Transformation which is involutive: when one twice applies it to a point or a figure the starting figure is found. Among current symmetries, one can quote the symmetry-mirror (reflection in a mirror) central symmetry, etc

A geometrical symmetry is a particular case of Symétrie. There exist several kinds of symmetries in the plan or space.

Symmetry in the plan

Symmetry compared to a point

Presentation

The symmetry of center O is the transformation which, with any point M, associates the point Me such as O is the medium of.


Construction: Plot the straight line (d) passing by has and O. Prolong it beyond O. With a compass pointed out of O and a spacing equal to OA, recut (d) in A'.

The only point invariant of this symmetry is the point O.

A symmetry of center O is also a rotation of flat angle and a homothety of center O and report/ratio -1

Center of symmetry

A figure has a center of symmetry C if it is invariant by the symmetry of center C.

Examples of center of symmetry:

  • letters NR, S and Z have a center of symmetry. Other letters have a center of symmetry because they have two axes of symmetry, one will discover them in the chapter of orthogonal symmetry in a plan.
  • a parallelogram has for center of symmetry the point of intersection of its diagonals. This property is characteristic of the parallelograms: a quadrilateral ABCD having this property is necessarily a parallelogram.
  • the Hexagone is a polygon which admits the intersection of its diagonals like center of symmetry.
  • the circle admits its center like center of symmetry.
  • analyzes some, a curve of equation there = F ( X ) has a center of symmetry C ( has ; B ) if and only if, for any reality H such as has + H belongs to the field of definition of F , one has
    • has - H belongs to the field of definition
    • F ( has + H ) + F ( has - H ) = 2 B
When the center of symmetry is at the origin of the reference mark, the function is known as odd. In this case the preceding expression is simplified in: F (- H ) = - F ( H )

Group symmetries power station-translations

The made up one of two symmetries of centers O and O', sO' O sO is a translation of vector 2 \ overrightarrow {OO'}


the theorem of the mediums makes it possible to notice that \ overrightarrow {MM } =2 \ overrightarrow {OO'}

This property makes it possible to define a first group of transformations of the plan: that of symmetries power station-translations. Indeed, by composing two central symmetries or translations, one obtains a central symmetry or a translation. And, to obtain the identical application, it is enough to compose a translation of vector U by the translation to vector - U , or to compose a central symmetry by itself.

Central symmetry preserves the directed distances and angles. It is thus a positive Isométrie or displacement. The group defined previously is thus a sub-group of the group of displacements.

Orthogonal symmetry compared to a line

Presentation

They are called also reflections of axis ( D ) . The reflection of axis ( D ) is the transformation of the plan which leaves all the points of ( D ) invariants and which, with any point M not located on ( D ), associates the point Me such as ( D ) that is to say the mediator of. As there exist two equivalent definitions of the mediator, one thus knows two equivalent constructions of the point Me.

Construction

Data: the axis of symmetry ( D ), the point has .

Objective: to build symmetrical A' of has by the orthogonal symmetry of axis ( D ).

  • First method:

Tracez a line perpendicular to ( D ) passing by has . This line cuts the axis in a point H .
With the compass pointed in H and drawn aside until has , to recut the line ( AH ) in A'

  • Second method:

the point B being given, one seeks the B' point such as the axis ( D ) must be the mediator of.

to build the B' point we will use the following property: Any point of a Médiatrice of a segment is equidistant ends of this segment .

We choose two unspecified points c1 and c2 of ( D ) and we will determine a B' point such as c1B=c1B' and c2B=c2B'.

Ainsi we are certain that ( c1c2 ), i.e. D , is the mediator of.

Choose c1 and c2 on ( D ).

Place the dry point of the compass on c1 and draw aside the other connects until B . Trace an arc.
Exécutez the same thing with the point dries in c2 .
the two arcs are cut out of B and B'.

Axis of symmetry

A figure has an axis of symmetry ( D ) if and only if it is invariant by the reflection of axis ( D )

Examples of usual figures:

  • the letters has, B, C, D, E, K, M, T, U, V, W have an axis of symmetry
  • the Cercle has a Infini t-pieces of axes of symmetry: all its Diamètre S.
  • a unspecified Angle always has an axis of symmetry: its Bisecting.
  • the Triangle Isocèle has an axis of symmetry: its Bisecting principal.
  • the equilateral triangle has 3 axes of symmetry: its 3 bisectrices.
  • the Losange has 2 of them: its 2 Diagonale S.
  • the Rectangle has 2 of them: its 2 Médiane S.
  • the Carré has 4 of them: its 2 diagonals (since it is also a rhombus) and its 2 medians (since it is also a rectangle).
  • analyzes some, a curve of equation there = F ( X ) has an axis of symmetry of equation X = has if and only if, for any reality H such as + H has belongs to the field of definition, one a:
    • has - H belongs to the field of definition
    • F ( has + H ) = F ( has - H )
When the axis of symmetry is the axis ( OY ), the function is known as pair

A figure having two perpendicular axes of symmetry has as a center of symmetry the point of intersection of the two lines. For example, the letters H, I, O, X thus have two perpendicular axes of symmetry a center of symmetry, in the same way the rectangle, the rhombus and the square.

Reflection and group of the isométries

The reflection preserves the distances and the angles. It is thus a Isométrie. But it does not preserve the orientation. It is said that it is one antidéplacement.

The made up one of two reflections of parallel axes is a translation.


the vectorial properties of the mediums make it possible to say that
\ overrightarrow {MM } = 2 \ overrightarrow {HH'} = 2 \ vec {U}

Made up of two reflections of the axes secant one is a rotation.


the properties on the bisectrices make it possible to say that
(\ overrightarrow {OM}, \ overrightarrow {OM }) = 2 (\ overrightarrow {OH}, \ overrightarrow {OH'}) = 2 \ alpha
It is noticed whereas the whole of the reflections generate all the whole of the plane isométries

Symmetry obliques

Symmetry compared to a line ( D ) according to a direction (of) (not parallel with ( D )) which is the transformation leaves all the points of ( D ) invariants and which, with any point M not located on ( D ) associates the point Me such as the line (ME) either parallel with (of) and medium of '' or on ( D )

This symmetry is quite involutive: the symmetrical one of is to Me well M . It offers less interest than her cousins because it does not preserve the distances: it deforms the figures. However, it preserves the barycentres and thus forms part of the transformations closely connected.

Symmetry in space

central Symmetry

One finds the same definition and the same properties as for central symmetry in the plan, with this close a central symmetry does not preserve the orientation in space.



The catch raises the right hand and its image raises the left hand.




Orthogonal symmetry compared to a line

One finds the same definition as in the plan. An orthogonal symmetry compared to a line is also a rotation of axis ( D ) and of flat angle.

Contrary to what occurs in the plan, such a symmetry in space preserves the orientation.


The catch raises the right hand and its image raises the right hand.


Orthogonal symmetry compared to a plan

Orthogonal symmetry compared to the plan ( P ) is the transformation which leaves all the points of ( P ) invariants and which, with any point M not located on ( P ), associates the point Me such as ( P ) that is to say the mediator plan of ''

Such a symmetry preserves the distances and the angles but does not preserve the orientation. This is why, when you raise the right hand in front of your mirror, your image raises its left hand.

It is shown that the whole of symmetries compared to plans generates by composition all the whole of the isométries of space

Oblique symmetries

One can just as easily define symmetries of axis ( D ) according to the direction ( P ) or of symmetries compared to ( P ) according to the direction ( D )

But these transformations are not isométries if ( D ) and ( P ) are not orthogonal. These transformations preserve the barycentres however and are particular cases of transformations closely connected of space.

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