Isometries of the line

James Parks

SUNY Potsdam

Abstract This is a prequel to [4]. It is a brief study of the isometries of a line, various groups of such isometries, and the topology of these groups.

Consider the set of all isometries of a line R to itself (we will identify the line with the set of reals R). This is the set of all transformations T:R - > R, such that if T(A) = C, and T(B) = D, for A,B in R, then |AB| = |CD| (as vectors).

The following 2 isometries of R are evident.

Let T_AB denote the translation of R by directed line segment (vector) AB in R, that is if T_AB(P) = Q, then AB = PQ (as vectors). We can always assume that T_AB = T_0D for some D in R, where 0D = AB. If AB = 0, then T₀ = i, where i denotes the identity isometry. Clearly translations are isometries and T_AB^-1 = T_BA.

Let R_P denote the reflection about point P, that is if R_P(A) = A', then P is the midpoint of line segment AA'. Reflections are clearly isometries and R_P^-1 = R_P, Figure 1.

Figure 1

We choose the positive orientation of a line segment in R to mean in the positive direction of R as a number line. Then translations T_AB are seen to be orientation preserving, while reflections in points R_P are orientation reversing.

It is well known that all isometries of Rⁿ are the composition of at most n+1 reflections of Rⁿ [1] [2], so we know that any isometry of R is the composition of at most 2 reflections.

The following compositions of combinations of the reflections and translations above will determine all possible isometries of R.

First, the composition of 2 reflections is a translation, R_AR_B = T_CD, where CD is in the direction of AB, and |CD| = 2|AB|.

The composition of 2 translations is a clearly a translation, T_CDT_AB = T_EF, where EF = AB + CD (as vectors).

The composition of a reflection and a translation, in any order, is a reflection:

1. R_PT_AB = R_P', where P' = T_CD(P), for CD in the direction of –AB, and 2|CD| = |BA|, Figure 2A.

2. T_ABR_P = R_P', where P' = T_EF(P), for EF in the direction of AB, and 2|EF| = |AB|, Figure 2B.

Figures 2A, 2B

Therefore the only types of isometries of R are reflections about points R_P and translations T_AB.

Some sets of isometries of R form algebraic groups under certain restrictions or choices.

The set of all isometries of R is a group under composition, denoted Iso(R), since the composition of 2 such isometries is an isometry, i is an isometry, and all such isometries are invertible with inverses which are isometries. We always assume the affine topology on Iso(R), hence Iso(R) is a topological group [2].

Reflections R_P form 2-element subgroups of Iso(R), <{R_P, i}, ^. >, since R_PR_P = i (so R_P^-1 = R_P).

Let Trans(R) denote the set of all translations of R, Trans(R) = {T_AB | AB in R}. Then, by the above observation on compositions of translations, and the fact that T_AB^-1 = T_BA and T₀ = i, Trans(R) is a (topological) group under composition.

Observe that Trans(R) is isomorphic to R, Trans(R) ~ R, by the isomorphism T_AB -> P, where AB = OP as vectors [1], [2].

Let Pres(R) denote the set of orientation preserving isometries of R. Then Pres(R) is a subgroup of Iso(R), since the composition of any 2 orientation preserving isometries of R is orientation preserving and the inverse of an orientation preserving isometry is orientation preserving. Also, Pres(R) = Trans(R), since translations are the only orientation preserving isometries of R from the calculations above. Thus we have Pres(R) ~ R.

The subgroup Pres(R) of Iso(R) is normal in Iso(R), Pres(R)<Iso(R), for given reflection R_P and translation T_AB, it follows that R_PT_ABR_P ^-1 = T_CD, where CD = R_P(AB) = -AB = BA.

Thus the quotient group Iso(R)/Pres(R) = {Pres(R), Rev(R)}, where Rev(R) = R_PPres(R) denotes the coset of orientation reversing isometries. Observe that given a reflection R_Q there exist translations T_PQ, T_QP such that T_PQR_PT_QP = R_Q.

Since the only orientation reversing isometries of R are reflections, we have the coset Rev(R) = {R_P | R in R}. Although it is not a subgroup of Iso(R), Rev(R) is homeomorphic to R, as an affine space, Rev(R) ~ R, by the map R_P -> P, cf. [2].

In general, Iso(Rⁿ) is isomorphic to the semidirect product of the translation group of Rⁿ, Trans(Rⁿ), with the orthogonal group of degree n, O(n), thus Iso(Rⁿ) ~ Trans(Rⁿ) ⋊ O(n) ~ Rⁿ ⋊ O(n) [1], [2], [3]. Since O(n) is the union of 2 copies of the special orthogonal group SO(n), we know that, topologically, the isometry group of Rⁿ is homeomorphic to 2 copies of RⁿxSO(n), Iso(Rⁿ) ~ Rⁿ x SO(n) U Rⁿ x SO(n).

For the dimension 1 case Iso(R¹) ~ Trans(R) ⋊ O(1) ~ R¹ ⋊ O(1) ~ R ⋊ Z₂ ~ (-1, +1)U(-1, +1), since O(1) ~ Z₂ ~ {-1, +1}, SO(1) ~ {+1}, and R ~ (-1, +1), Figure 3.

Figure 3

There are many other subgroups of Iso(R). For example consider the subgroup H generated by the single reflection R₀ and a single translation T_AB (we assume AB has positive orientation). This subgroup will contain all translations T_k0C, where k in Z, 0C = AB, k0C is in the direction of AB if k>0, and in the direction of –AB = BA if k<0, and |k0C| = |k||0C|. Observe that T_k0C^-1 = T_-k0C. This subgroup will also contain all reflections R_Q, where 0Q = k0P = (k/2)AB, for some k in Z, and P is the midpoint of XX', where T_ABR₀(X) = X', X arbitrary, X not 0, then repeat this composition k times, Figure 4. Note that the composition of 2 such reflections is a translation of the above type, R_PR_Q = T_0D, where 0D = hAB, for some h in Z. Also, the composition of a reflection R_P and a translation T_DE in H is a reflection T_DER_P = R_Q, for Q the midpoint of YY', where T_DER_P(Y) = Y', for Y arbitrary, Y not 0, and 0Q = (j/2)AB for some j in Z.

Figure 4

The collection of all of the above translations {T_k0C : k in Z, and 0C = AB}, is obviously a subgroup of H we denote by Trans(AB). It forms a normal subgroup of H, Trans(AB)<H, since R_PT_k0CR_P^-1 = T_k0C^-1 = T_-k0C, for all T_k0C in Trans(AB), R_P in H. We also have Trans(AB) isomorphic to Z, Trans(AB) ~ Z, by the isomorphism k0C -> k.

The subgroup <{R₀, i}, ^. > of H is not normal in H, since T_k0CR₀T_k0C^-1 = R_P for 0P = k0C, k in Z, and 0C = AB.

Every element of H has the form T_k0CR_P, for some k in Z, since R_PT_k0C = T_k0C^-1R_P by the above observations on the normality of Trans(AB) in H.

Thus H = Trans(AB) ⋊ <{R₀, i}, ^. > ~ Z ⋊ Z₂.

Topologically, this example will consist of the set of points nC/(1+|n|), n in Z, on the Pres(R) line in Figure 3, and the set of points nP/(1+|n|) = (n/2)C/(1+|n|), n in Z, on the Rev(R) line in Figure 3, see Figure 5.

Figure 5

It may be assumed that the subgroup generated by a fixed reflection R_P and all translations {T_AB} is a proper subgroup of Iso(R), as is the case for isometries of the plane R² [4]. However this subgroup is actually all of Iso(R), since all reflections are generated by the set of a fixed reflection R_P and all translations {T_k0C}, as noted in the discussion of the quotient group Iso(R)/Pres(R) above.

References

1. G. Martin, Transformation Geometry, 4^th printing, S-V, 1982.
2. E. Rees, Notes on Geometry, 2^nd edition, S-V, 1988.
3. D. Dummit and R. Foote, Abstract Algebra, 3^rd ed., John Wiley & Sons, 2003.
4. J. Parks, Subgroups of the Group of Isometries of the Plane, J. Geom. & Topo., 7(3)(07), 405-422.