What is the group of isometrics

Permutations and isometries

For every set M, the set of all bijections from M to M together with the composition forms a group.

definition(Permutation group or symmetry group)

Let M be a set. We sit:

๐’ฎM. = {f: M โ†’ M | f is bijective}.

โŒฉโ€ฏ๐’ฎM., โˆ˜ใ€‰ or ๐’ฎ for shortM. is called the Permutation group on M.

Every subgroup of a permutation group leads to a natural notion of equivalence:

definition(Equivalence with respect to a subgroup)

Let M be a set and let ๐’ข be a subgroup of ๐’ฎM..

We define for A, B โŠ† M:

A โˆผ๐’ข B.if an f โˆˆ ๐’ข exists with B = f โ€ณ A.

If A โˆผ applies๐’ข B is the name of A and B. ๐’ข-equivalent or ๐’ข-like.

exercise

โˆผ๐’ข is an equivalence relation.

In accordance with the Erlangen program, the reduction of the full permutation group of a set M to a - geometrically motivated - subgroup creates a "geometry on M" with an equivalence term for "figures" in M. An obvious goal of the investigation is now a as precise an understanding as possible of the subgroup under consideration. We want such an investigation for the group of distance-maintaining bijections of the Euclidean spaces M = โ„, โ„2, โ„3 carry out. This group is itself a subgroup of the Euclidean main group of conformal images. In general we define (for all n โˆˆ โ„•):

definition(Isometry, isometry group)

Let f: โ„nโ†’ โ„n bijective. f is called a (Euclidean) isometry(des, im) โ„n, if for all x, y โˆˆ โ„n applies:

d (f (x), f (y)) = d (x, y). (Distance fidelity)

We sit:

โ„n = {f โˆˆ ๐’ฎโ„n | f is an isometry}.

ใ€ˆโ„n, โˆ˜ใ€‰ or โ„ for shortn is also called the Isometric group of the โ„n.

Other geometrically motivated subgroups of ๐’ฎโ„n are about: The group of homeomorphisms f: โ„nโ†’ โ„n, the group of equidistant maps, of the orthogonal maps, and further the group of bijective affine maps. The concept of affine mapping leads to projective geometry by introducing an infinitely distant point.

Instead of โ€œf: โ„nโ†’ โ„n bijective โ€the requirementโ€œ f: โ„ is sufficientnโ†’ โ„n surjective โ€, since a function that is accurate in distance is automatically injective. Here the question arises whether one can even dispense with the requirement of surjectivity, i. H. the question: Is an equidistant mapping automatically surjective and therefore an isometry? We will see below that this is actually the case.

The isometric group allows the following simple definition of the well-known concept of congruence for figures in โ„n:

definition(Congruence)

Let A, B โŠ† โ„n. A and B are called congruent, in characters A โ‰ก B, if A โˆผโ„n B holds.

For the visual space, congruence is often explained as follows: A and B are congruent if A and B can be brought into congruence by simply or repeatedly moving, rotating and mirroring. This descriptive explanation of the concept of congruence is in fact consistent with the more abstract definition. To put it somewhat casually: There are no complicated, unreporting isometrics. In addition, the result will be that the composition of isometries does not go beyond very few, easy-to-describe basic types that only have to be carried out once and not repeatedly. Overall, we will provide a simple description of all isometrics of the โ„n for the first three dimensions. In general, such cataloging becomes more and more complex, the higher the dimension of the room. For the important special cases n โ‰ค 3, however, a surprisingly clear and beautiful characterization of all isometries is possible.

In our concept of congruence, reflections lead to straight lines in โ„2, Levels in the โ„3, etc. to congruent figures. The term thus particularly identifies certain figures of the โ„3that cannot be converted into one another by "real" movements. Further applies in โ„2 for example, that the letters โ€œpโ€ and โ€œqโ€ are congruent. A real overlay is possible here, but requires a detour via the third dimension.

Isometrics are also often referred to as rigid mappings. The choice of words is supported by the following observation:

sentence(Isometrics that fix the zero point are conformal)

Let f: โ„nโ†’ โ„n an isometry with f (0) = 0.

Then for all x, y โˆˆ โ„n:

w (f (x), f (y)) = w (x, y).

proof

The two triangles x, 0, y and f (x), 0, f (y) in โ„n have the same side lengths, since f is an isometry. The assertion follows from this through elementary argumentation.

For any isometry f: โ„nโ†’ โ„n it holds that the angles at y or at f (y) in all triangles x, y, z and f (x), f (y), f (z) coincide.

As a corollary we get that isometries f with f (0) = 0 also receive the scalar product: We have ใ€ˆf (x), f (y)ใ€‰ = ใ€ˆx, yใ€‰ for all x, y โˆˆ โ„n. Because ใ€ˆx, yใ€‰ ยท y is the projection of x onto y for all x, y โˆˆ โ„n. This projection depends only on the length of the two vectors and their angle. So the length of ใ€ˆx, yใ€‰ y is equal to the length of ใ€ˆf (x), f (y)ใ€‰ f (y), and so ใ€ˆx, yใ€‰ = ใ€ˆf (x) , f (y)ใ€‰. (See also the above cosine formula (c).)

Conversely, a mapping that contains scalar products is automatically an isometry f with f (0) = 0. We will prove this below.