Reflection (mathematics)
Reflections are in geometry certain Kongruenzabbildungen the plane or the ( Euclidean ) space. A glide reflection is the combination of a reflection and a translation. There are also inclined mirrors which are not Kongruenzabbildungen.
Point reflection
It is a figure that is given by a point Z ( mirror point, center ). The reflection at the point Z is arranged each point P of the plane or space an image point P ', to which is determined by the fact that the link [ PP' ] is halved from point Z.
Sometimes the point of reflection is also referred to as a room mirror, or inversion; However, note that the term inversion is often also used for another picture, the reflection in a circle.
A point reflection has exactly one fixed point ( ie a point that the picture can be changed), namely the center Z. Fixgeraden (ie, the straight line which transfers the image in itself) are precisely the lines through Z. An arbitrary line g is ' mapped to a straight line parallel to g (Fig. line) g.
In the plane, the point reflection at the center Z is equivalent to a rotation of 180 ° around the center of rotation Z.
Point reflections are straight, length and angle preserving, so Kongruenzabbildungen.
Any flat point reflection can be replaced by two consecutively executed axis reflections, the axes of these reflections go through the center and Z are perpendicular to each other. The order of these reflections is of no importance.
Each spatial point reflection can be replaced by three consecutively performed plane reflections, the three mirror planes pass through the center and Z are perpendicular to each other. The order of these reflections is of no importance.
In crystallography, a point reflection inversion or dot inversion center is named and marked with the Hermann- Mauguin symbol 1.
See also: point symmetry
Synthetic geometry
In the synthetic geometry, a point reflection in every affine translation plane which satisfies the ( affine ) Fano axiom can be defined. To this end, the center of the reflection point is determined chosen as the origin and reversible uniquely assigned to each point of the plane, the translation as a local " vector ". The point reflection is defined by and we have:
- At each point there is exactly one point reflection at this point,
- Each point reflection is a partial relationship loyal collineation, ie an affinity
- Each point reflection is in involution,
- The point reflection on induced by the truer -1 endomorphism of the translation group and therefore a central dilation (→ see dilation ) with the elongation factor -1,
- In a präeuklidischen level include the reflections point to the Kongruenzabbildungen.
→ Refer to the used generalized terms the article " affine translation plane " for a definition of point reflections in arbitrary affine planes, which generalizes the definition given here, the article " Fano axiom ".
Mirror image
A mirror image (even straight mirroring) is given by a straight line a ( mirror axis or short axis). It assigns to each point P, an image point P ' to which is determined by the fact that the link [ PP' ] of the axis A at right angles is halved.
The fixed points of a mirror image are exactly the points of a One therefore speaks of the fixed point of a straight line The Fixgeraden the mirror image are exactly the axis a perpendicular straight line to the axis itself and all. In the spatial case, there is also Fixe planes, namely the axis a orthogonal planes.
The mirror image is a congruence.
If two congruent objects contained in the plane, this in any case by composition (concatenation, sequential execution ) can be converted to a maximum of three axis mirroring each other. The mirror image can therefore be used as a basic concept of metric geometry of the plane.
In the plane it should be noted that the orientation ( the machining direction ) is changed by a triangle one of the mirror. She's here so no actual " movement ", that is, they can not be realized by a physical movement without the object to leave the plane.
In three-dimensional space, the mirror image corresponds to a rotation by 180 ° around the axis of reflection. An object that is located together with the mirror axis in a plane, this " flipped" in the same plane; This is the movement that was not possible with the constraint on a plane.
Synthetic geometry
In the synthetic geometry is defined more generally a (vertical ) axis mirroring for general affine planes präeuklidischen levels. Here we mean by the reflection on the straight line ( the axis) that mapping of the plane on to which each point associates that point, which lies on the perpendicular straight line to pass, and is determined by the fact that the intersection of perpendicular straight line with the center of. Compare the figure to the right: the angle is a right, the vectors indicated and are inverse to each other, that is, the midpoint of the segment. This causes the image of under the mirror image of clearly defined.
For this vertical axis reflections applies:
Plane reflection
This other kind of mirroring occurs only in the room geometry. It is given by a plane α, the mirror plane. The pixel P is determined that the link between it and its image point P ' on the mirror plane at right angles is halved.
Fixed points are exactly the points of the mirror plane. Fixgeraden the straight line of the mirror plane and the straight line orthogonal to this. Fixed planes are the mirror plane and orthogonal to their planes.
The plane reflection changed the orientation of a simplex '. So, too, is not a " real" movement: A tetrahedron can not be physically converted into its mirror image.
In crystallography mirroring is designated by the Hermann- Mauguin symbol m.
Reflections in spaces of arbitrary dimension
In an n- dimensional Euclidean space, there are n types of reflections, namely reflections to 0, 1, ... (n-1 )-dimensional subspaces ( mirror elements ).
Fixed points are always the points of the mirror element. Higher-Dimensional fixed elements are the subspaces and the subspaces that are orthogonal to this.
The reflection of an (n -1 )-dimensional subspace can not each a " real movement " understood in n- dimensional space. When embedded in an (n 1) -dimensional space it is equivalent to a self-reciprocal rotation around the mirror element.
That implies amongst other things, that in the one-dimensional case (ie on a straight line ) the point reflection is the only possible reflection, and that this, as it reverses the order of the points that can not be understood as a movement without leaving the line.