Congruence mapping

A rectangle and a right-angled triangle (1) with three figures congruent to the original figure; the mediating congruence maps are
axis reflection (2)
displacement (3)
rotation (4)

In elementary geometry , synthetic geometry and also in absolute geometry, a congruence mapping (from the Latin congruens ,  matching '' ) is a geometric mapping in which the shape and size of any geometric figures are not changed, i.e. every figure is mapped to a congruent one . In particular, congruence maps leave the distance between any two points unchanged (invariant). The terms "congruence mapping" and " movement " are synonymous for Euclidean geometry , whereby mostly only plane movements are referred to as "congruence mapping". The more general meaning of the term in absolute geometry is indicated in the Synthetic and Absolute Geometry section of this article.

Appearance and properties

Congruence mappings in the plane of the drawing, i.e. in the Euclidean plane , are bijective mappings of this plane of the drawing that can always be put together by executing one behind the other (chaining, composition) of axis reflections .

A distinction is made between actual and improper congruence maps. The actual congruence maps are characterized by the fact that they can be displayed by linking an even number of axis reflections , while the improper congruence maps require an odd number . It has been proven that a display with a maximum of three interlinked axis reflections is always possible. If a congruence mapping has a fixed point , it can be represented by linking (at most) two axis reflections.

Congruence maps are straight, length and angle true. So you map straight lines on straight lines and leave path lengths and angle sizes unchanged. They are also bijective , that is, reversible, and their inverse maps are always also congruence maps.

The set of plane congruence maps consists of

From an algebraic point of view, the congruence maps of the drawing plane form a group . Congruence maps are special similarity maps , even more generally they belong to the affinities .

In analytical geometry , (plane) congruence maps are described with the help of matrices , and they make up the Euclidean group . ${\ displaystyle {\ rm {E (2)}}}$

Geometric construction in the drawing plane

The general case

Example for the construction of a congruence map, here a rotation

For two given, congruent triangles and (in the case of congruence, the point is assigned to the point , etc.), a clearly determined congruence mapping can always be constructed. Compare the figure on the right: The two triangles are given, the corner points of which are connected by red and green lines. ${\ displaystyle A_ {0} B_ {0} C_ {0}}$${\ displaystyle A_ {2} B_ {2} C_ {2}}$${\ displaystyle A_ {0}}$${\ displaystyle A_ {2}}$

1. Construct the perpendicular of the line .${\ displaystyle a_ {1}}$${\ displaystyle A_ {0} A_ {2}}$
2. Construct the mirror points of and ( or ) when mirroring on the axis . In the special case that is, the first two steps are omitted, then you put and .${\ displaystyle B_ {0}}$${\ displaystyle C_ {0}}$${\ displaystyle B_ {1}}$${\ displaystyle C_ {1}}$${\ displaystyle \ sigma _ {1}}$${\ displaystyle a_ {1}}$${\ displaystyle A_ {0} = A_ {2}}$${\ displaystyle B_ {1} = B_ {0}}$${\ displaystyle C_ {1} = C_ {0}}$
3. Construct the perpendicular of the line . Since the triangle is congruent to the construction and to the prerequisite to , the lines and are of equal length, the triangle is isosceles and goes through .${\ displaystyle a_ {2}}$${\ displaystyle B_ {1} B_ {2}}$${\ displaystyle A_ {0} B_ {0} C_ {0}}$${\ displaystyle A_ {2} B_ {1} C_ {1}}$${\ displaystyle A_ {2} B_ {2} C_ {2}}$${\ displaystyle A_ {2} B_ {1}}$${\ displaystyle A_ {2} B_ {2}}$${\ displaystyle A_ {2} B_ {1} B_ {2}}$${\ displaystyle a_ {2}}$${\ displaystyle A_ {2}}$
4. Construct the mirror point of when mirroring on . In the special case that is, the 3rd and 4th step are omitted and you bet .${\ displaystyle C_ {1} '}$${\ displaystyle C_ {1}}$${\ displaystyle \ sigma _ {2}}$${\ displaystyle a_ {2}}$${\ displaystyle B_ {1} = B_ {2}}$${\ displaystyle C_ {1} '= C_ {1}}$
5. Now two cases are possible: Either there is already considered , as in the example shown, the congruence can be represented by the composition of the said axis reflections or you have to have a reflection on the connecting line perform then by assumption to reflect. (Here the congruence theorem SSS is used in the form: If two different points and the plane are given, then there are at most two points , so that the line lengths and satisfy with fixed numbers . If there are two points with these properties, then the triangle goes by mirroring the axis on the connecting straight line into the triangle .)${\ displaystyle C_ {1} '= C_ {2}}$ ${\ displaystyle A_ {2} B_ {2}}$${\ displaystyle C_ {1} '}$${\ displaystyle C_ {2}}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle C}$${\ displaystyle {\ overline {AC}} = b}$${\ displaystyle {\ overline {BC}} = a}$${\ displaystyle a, b}$${\ displaystyle C, C '}$${\ displaystyle ABC}$${\ displaystyle AB}$${\ displaystyle ABC '}$
Chaining two reflections on parallel axes results in a shift; In addition, it is shown here for the first mirroring how the mirror image can be constructed with the help of two circles around any auxiliary points on the axis${\ displaystyle A_ {0} \ mapsto A_ {1}}$

Apart from the two special cases mentioned in 2. and 4., i.e. in those cases in which two different axis reflections are involved, the following distinction is interesting: ${\ displaystyle \ sigma _ {1}, \ sigma _ {2}}$

1. If the two axes of the reflections intersect at one point , the image is a rotation around and the angle of rotation of the rotation is twice as large as an oriented angle between the two axes. An oriented angle is thereby rotation angle of one of the two rotations , the on mapping which of the rotations is selected to indifferent.${\ displaystyle Z}$${\ displaystyle \ sigma _ {2} \ circ \ sigma _ {1}}$${\ displaystyle Z}$${\ displaystyle \ alpha = <\! \! \!) \, (a_ {1}, a_ {2})}$${\ displaystyle \ alpha}$${\ displaystyle Z}$${\ displaystyle a_ {1}}$${\ displaystyle a_ {2}}$
2. If the two axes are parallel, then the image displacement by a displacement vector whose length is twice the distance of the two axes and has the same direction as the perpendicular to the axes of displacement which on shifts, the picture compare right: It is obviously .${\ displaystyle \ sigma _ {2} \ circ \ sigma _ {1}}$${\ displaystyle {\ vec {v}}}$${\ displaystyle a_ {1}}$${\ displaystyle a_ {2}}$${\ displaystyle {\ overrightarrow {A_ {0} A_ {1}}} / 2 + {\ overrightarrow {A_ {1} A_ {2}}} / 2 = {\ vec {v}}}$

In addition, it emerges from the construction text that the congruence mapping between two planar, congruent figures is clearly defined by three pairs of points and image points, provided the three selected original image points do not lie on a common straight line. In other words: The group of congruence maps operates sharply simply transitive on each congruence class of triangles (understood as ordered point triples). This is the substantive reason why the congruence theorems for triangles play a dominant role in elementary geometry .

Glide and reflection

If two triangles are congruent in opposite directions , then they can always be mapped onto one another by gliding reflection

Two given, oppositely congruent triangles and (in the case of congruence, the point is assigned to the point , etc.) can always be mapped onto each other by gliding reflection , compare the figure on the right: ${\ displaystyle A_ {0} B_ {0} C_ {0}}$${\ displaystyle A_ {2} B_ {2} C_ {2}}$${\ displaystyle A_ {0}}$${\ displaystyle A_ {2}}$

1. Move the triangle (red) with the displacement defined by onto the (in the same direction congruent) triangle . In the special case , this shift is the identical mapping .${\ displaystyle A_ {0} B_ {0} C_ {0}}$${\ displaystyle {\ overrightarrow {A_ {0} A_ {2}}}}$${\ displaystyle A_ {2} B_ {1} C_ {1}}$${\ displaystyle A_ {0} = A_ {2}}$
2. Construct the vertical line of the line (green). As in the 3rd step of the general construction, you can see that this vertical line must also go through .${\ displaystyle B_ {1} B_ {2}}$${\ displaystyle A_ {2}}$
3. Reflect the triangle on the vertical line from the 2nd step, this results in a triangle . From the congruence theorem SSS (see above in the 5th step of the general construction) and the requirement that the given triangles are congruent in opposite directions, it follows . In the special case that is, the 2nd step is omitted and the 3rd step is mirrored on the connecting line . In this special case, this means that it is mapped to, for the same reasons as in the case of mirroring at the center perpendicular in the general case.${\ displaystyle A_ {2} B_ {1} C_ {1}}$${\ displaystyle A_ {2} B_ {2} C_ {1} '}$${\ displaystyle C_ {1} '= C_ {2}}$${\ displaystyle B_ {1} = B_ {2}}$${\ displaystyle A_ {2} B_ {2}}$${\ displaystyle C_ {1}}$${\ displaystyle C_ {2}}$

From this construction text follows:

• Two oppositely congruent triangles can always be mapped onto one another by means of a sliding reflection.
• The shift is then certainly the identical mapping, that is, the mapping is a pure reflection, if at least one pair of assigned corner points assigns a point on the plane to itself in the case of congruence.
• If and only if one of the routes between associated points , , the length 0 has or collapse the center perpendicular of all three routes that glide is a pure reflection.${\ displaystyle A_ {0} A_ {2}}$${\ displaystyle B_ {0} B_ {2}}$${\ displaystyle C_ {0} C_ {2}}$
• If two triangles are congruent in opposite directions and if they match in at least one corner point that is assigned to themselves, then they match in two such corner points or the perpendiculars of the other two lines between assigned points coincide. This is a generalization of the above formulation of the congruence theorem SSS.

Together with the general construction text, this results in:

• A non-identical rotation can always be represented as a composition of two different axis reflections. These axis reflections are generally not clearly determined by the rotation, but their axes always intersect at the center of the rotation and the relationship to the angle of rotation described above applies to the angle between the axes.
• Executing a rotation and an axis mirroring in any order is always a sliding mirroring.

• Executing a shift and a non-identical rotation in any order is a rotation.
• In general, a glide mirror - moving around a vector and then mirroring on an axis - becomes a different glide mirror if you swap the order of the two congruence maps.${\ displaystyle {\ vec {v}}}$${\ displaystyle a}$

Differentiation from the movements

From the standpoint of analytic geometry, there is no difference between the terms congruence mapping and motion . In elementary geometry, however, only one movement of the Euclidean plane is usually called congruence mapping , i.e. only movements in the two-dimensional case .

School math

School mathematics is initially based on the plane of the drawing. The term "congruence" is made tangible for simple figures (triangles, squares, circles, ...):

• You can cut out a figure, which is ideally drawn on translucent paper or foil, and place it on top of the other figure in such a way that both “come together”, so the two figures are “congruent”.
• You can draw two congruent figures with one and the same “template”.

Building on this, the later experience is that you can “sometimes” see the congruence of figures by drawing without cutting them out.

1. Sometimes you can make two figures drawn on the same translucent sheet of paper coincide by folding the paper without cutting them out. The folding can then be abstracted to construct an axis reflection .
2. Instead of actually cutting out a figure and shifting it “without twisting” it, the shift can be shown abstractly using parallel arrows of equal length at the corners of a figure .
3. Similarly, a real, physical rotation of a cut-out figure, which is pinned to an inner point with a needle, can be represented in an abstract drawing with the circles of motion of the corner points.

The fact that with these three graphic abstractions and their combinations one can replace all “experimental” proofs of congruence by cutting out and superimposing them should be grasped intuitively. (In mathematical terms: that the group of congruence mappings is generated by mirror images, shifts and rotations.) However, it is proven that each shift and each rotation can be replaced by two suitable mirror images.

An example of the problems that arise in school mathematics is the demarcation between “equality” of geometric objects and “congruence”: Intuitively and colloquially, two equally long, i.e. congruent routes are viewed and designated as “the same”. The student's idea that there is no need to distinguish between identical congruence and identity of figures poses a didactic problem.

Some problems can be countered by exemplarily “real deductions”, ie proofs, in which the congruent figures cannot be identical due to the problem or in which the order of the assigned points is important for congruence.

Synthetic and absolute geometry

In the axiomatic structure of a Euclidean plane in synthetic geometry and in the structure of absolute geometry, the terms "congruence mapping" and "plane motion" belong to different deductive approaches:

1. From congruence to congruence mapping : One describes the congruence as a basic term axiomatically. In Hilbert's system of axioms of Euclidean geometry , this is realized by two equivalence relations on the set of lines and angles of the plane (group III, axioms of congruence). Then a congruence mapping is a bijective self-mapping of the level in which each congruence class is mapped to itself.
2. From the concept of reflection or the group of movements to congruence : the plane is equipped with an axiomatically defined orthogonality relation. This then enables the definition of vertical axis reflections, see pre-Euclidean plane . A movement of the plane is then every image that can be represented as a composition of a finite number of such axis reflections. Another related approach, also used in absolute geometry , axiomatically defines the properties of the motion group itself. In both approaches, the movement group is fundamental and congruence is a term derived from it. Two figures are congruent if they belong to the same trajectory of the group operation in which the motion group operates on the set of all subsets of the plane ( power set ).

School geometry in Germany is generally based on a deductive approach according to Hilbert, in which congruence is a basic concept. In the plane one speaks only of congruence mapping and the term movement occurs, if at all, only in the three-dimensional geometry.

literature

• HSM Coxeter : Immortal Geometry . Birkhäuser Verlag, Basel / Stuttgart 1963 (translated into German by JJ Burckhardt).
• Hans Schupp: Elementary Geometry . Schöningh, Hannover 1977, ISBN 3-506-99189-2 .

Congruence in school mathematics:

• Marianne Franke: Didactics of Geometry . Ed .: Friedhelm Padberg. reprinted. Spectrum, Academic Publishing House, Heidelberg / Berlin 2000, ISBN 3-8274-0994-2 .

Synthetic geometry:

In addition to a deductive structure of geometry, the following books also provide tips for application in the didactics of geometry for high schools:

• Wendelin Degen, Lothar Profke: Fundamentals of affine and Euclidean geometry . Teubner, Stuttgart 1976, ISBN 3-519-02751-8 .
• Günter Pickert : Deductive geometry in high school lessons . In: Mathematical semester reports . tape X . Springer, 1964, p. 202-223 .
• Lothar Profke: From affine to Euclidean geometry with the help of an orthogonality relation . In: Der Mathematikunterricht 22: 4 (axiomatics of affine and Euclidean planes), 36-86 . Friedrich, Hannover 1976, p. 36-86 .