Affine_Reflexion

1. Affine reflexion

Affine reflexions represent the simplest kind of affinities and also special cases of affine axial homologies (see Affinity.html and Affinity_Fixed_Point.html ). They generalize the euclidean reflexions and are defined by giving two lines {L, L'}. L is called the axis and L' is the conjugate direction. The corresponding transformation often denoted by f = (L,L') associates to each point P of the plane a point P', such that PP' is parallel to L' and its middle M is on line L.

The difference to the usual euclidean reflexion is that the "mirror" L is now not necessarily orthogonal to the conjugate direction L'.
Following properties are immediate consequences of the definition:
1) Affine reflections leave all points of the "mirror" L fixed.
2) They leave also invariant all lines L'' parallel to L'.
3) They map every line parallel to the mirror L to its symmetric parallel with respect to L.
4) They are involutive maps i.e. their inverse equals to the map itself or f² = Id (twice applying the map gives the identity).

2. Involutive affinities

Next proposition is a general characterization of the affine reflexion related to its property (4) above.
A different from the identity involutive affinity F (i.e. satisfying F² = Id) is either a symmetry at a point (called often half-turn) or an affine reflexion.

The clue here is the consideration of an arbitrary point X, its image X' and the line N of these two points. Obviously N is invariant by F and there are two cases.
1) All lines N=XX' for all possible points are parallel or identical.
2) There are two points {X,Y} for which XX' and YY' intersect at a point O.
In both cases the middle M of XX' is mapped to itself, since M'=F(M) is also a point of the line XX' and by the preservation of ratios by affinities M'X'/M'X = MX/MX' = -1.
Thus, in the first case the definition of the affine reflexion is satisfied. The line L' is determined by the common direction of XX' and line L is determined by two points {X,Y} and their images {X',Y'} by taking the middles {O,O'} of segments {XX', YY'} respectively. In the second case O is obviously a fixed point of F and it is the common middle of XX' and YY' for the same reason as before.

Thus, XYX'Y' is a parallelogram and by a ratio-argument again we see easily that each point P of XY maps to the O-symmetric point P' of X'Y'. The proof is completed by showing (using again preservation of ratios) that each point on OP maps to its O-symmetric on OP'.

3. Translations

The product f₂*f₁ of two reflexions f₁ = (L₁, V) and f₂ = (L₂, V) with parallel axes L₁, L₂ and common conjugate direction is a translation i.e. a map defined by a fixed vector v which to every point P corresponds P' = P + v.

The proof follows from the picture. F₁(A) = B, F₂(B)=C implies that AC is the double of DE i.e. the distance of L₁, L₂ in the direction V. This defines a vector of constantdirection (that of V) and length (v = AC is a free vector of fixed length and direction), hence shows the claim.
Remark-1 Notice that the relevant ingredient in the representation of the translation as a product of reflexions is the conjugate direction. In fact we can "turn" the two parallels{L₁, L₂} to a new position taking care only that their distance in the V-direction remains the same and equal to half the measure of v. Then the translation can be again represented as a product of thetwo reflexions with respect to the two parallel lines in their new position.
Remark-2 Translations build a subgroup of the group of all affinities of the plane. The composition f₂*f₁ of two translations by the vectors v₁, v₂ is the translation by the vector v₁+v₂.This is not so for the set of reflexions. The product of two reflexions is not a reflexion. The previous representation of a translation as a product of two reflexions gives a counterexample.

4. Shears

Shears are products s=g*f of two affine reflexions {f,g} whose "mirrors" are identical. The figure below shows the image s(P)=g(f(P)) of a typical point P under such a composition of transformations. Line L is the common mirror of the two reflexions, segment AB defines the conjugate direction of f and segment BC defines that of g.

The basic observation is that triangle PP'P" with vertices {P, P'=f(P), P''=g(P')} has fixed angles i.e. the triangles resulting for various positions of P are all similar to each other. Thus, also the direction of the median MP'' is fixed and all triangles PMP'' are also similar to each other. This enables a quick construction of P'' once the direction of the median MP'' has been determined:
1) Project P on the first mirror L along the conjugate direction L' of f to the point M.
2) Draw from M parallel to the direction of the median and find its intersection P'' with the parallel from P to the mirror L.
The points of the mirror L are left fixed by the shear. Besides each parallel to L is left invariant by the shear which, restricted there, coincides with a translation. Thus, moving P on the parallel to L through that point gives a point P''=s(P) such that PP'' is parallel to L and has a constant measure (double that of MN).
Remark The product representing the shear s=g*f is not commutative g*f is instead the inverse of f*g. This follows from the elementary properties of the parallelogram:

5. Strains

If an affinity f fixes a line L then it is either a shear or a strain. Later is transformation somewhat more general than an affine reflexion. It is defined by giving the line L and two points {P, f(P)}.

Assume that PP' is not parallel to L and consider the intersection point Q of line PP' with the axis. Notice first that line PP' is invariant i.e. maps to itself under f. Then, for any point X and its image X'=f(X) draw the parallel XY to PP' intersecting L at Y. By the preservation of parallels by affinities and the constancy of Y follows that line XY is left invariant by f, hence X' is on XY. It is then easy to see that PX and P'X' intersect at a point Z of L and X'Y/XY = QP'/QP is constant. Thus the image X' is found in this case by a simple recipe:
1) Draw from X a parallel to PP'.
2) Take X' on this parallel so that YX'/YX = QP'/QP, where Y is the intersection point of XX' with L.
L is again called the axis of the strain and the direction of PP' (or any other XX') is called the conjugate direction of the strain. The constant ratio k = YX'/YX is called the ratio of the strain.
Remark-1 An affine reflexion is a special strain for which the ratio k = -1.
Remark-2 The excluded case in which PP' is parallel to L is the case of a shear (see section-5 of Affinity_Fixed_Point.html ). It follows that the only affine transformations which leave a single line L fixed are shears and strains having L for axis.

6. Conjugation by translations

Let f = (L,V) be a reflexion and t a translation defined by the vector v. Then the conjugate transformation f' = t*f*t^-1 is the reflexion
f' = (t(L),V).

Proof by picture. Y = t(X), Y'=t(X') and the correspondence of Y' to Y coincides with the affine reflexion on t(L). The conjugate directions of f and f' are the same.

7. Conjugation by reflexions

Next figure displays the proof of the more fundamental fact that the conjugate f' = g*f*g^-1 = g*f*g of a reflexion by another is again a reflexion. Reflexion f is described
as f = (L', V'), g = (L,V) and f'= (L'', V'').

The proof follows by observing that for an arbitrary point X its image C=f(X) the intersection of XC with L and the projection B of C on L along the direction of V there is defined a triangle ABC. This triangle has an invariant similarity type for all positions of X, hence the triangle CAD resulting by extending to its double the basis CB has also a constant similarity type. Take then D = g(C), X'=g(X) and {M,N} respectively the middles of {XC, DX'}. Triangles MAN resulting for the various positions of X are all similar to each other and their vertices {A,M} move on the two fixed lines {L, L'}, hence its other vertex N will vary also on a fixed line L''. Obviously the correspondence of X' to D defines the affine reflexion f' = (L'', V'') = g*f*g = g*f*g^-1.
Remark-1 Since translations are products of two reflexions this result implies the result of the previous section.
Remark-2 The conjugate direction V'' of f' is the harmonic conjugate of AC with respect to the pair (AB, AW), where AW is the parallel to direction V from A.
Similarly the axis L'' of f' is the harmonic conjugate of L' with respect to the pair of lines (L, OU) later being the parallel to V from O.

8. Reflexion and translation (glide reflexions)

The product s = t*f of a reflexion and a translation can be further analyzed. The analysis depends on the relative position of the three elements determining the two factors f = (L,V) and the translation vector (t). The general case is that for which these directions are independent (parallel to the sides of a triangle).

Then according to section-3 we can write the translation t = f''*f' as a product of two reflexions , where the axes of {f',f''} are parallel to L. We can even translate {L₁,L₂} so that L₁ coincides with L. Thus, in this case the original transformation s = t*f can be written also in the form f₂* f₁*f, where {f₁,f₂} have the same conjugate direction with the one of {f', f''} and the axis of f₁ coincides with that of f.

The figure above shows the resulting configuration for the representation of s = f₂*f₁*f and the images of a point X under the various transformations.
X₀ = f(X), X' = f₁(X₀) = f₁(f(X)), X'' = f₂(X') = f₂(F₁(f(X))) = s(X).
It is easily verified that in this case the transformation can be written as a composition of the affine reflexion f₃ = (L₂, V), mapping X to X₁ and the translation X₁ to X''. This being parallel to L₂ and the length X₁X'' being constant, since triangle X₀X₁X'' is always similar to itself and X₀X'' has the length of the original translation vector. Such compositions of a reflexion plus one translation in direction parallel to the axis of reflexion are called affine glide reflexions. These are their properties
1) Line L₂ maps to itself. S restricted to L₂ coincides with a translation (thus has no fixed points on L₂).
2) Every line N parallel to L₂ (or/and L) maps to the line N' which is symmetric to N with respect to L₂.
3) The transformation s has no fixed points at all if the translation vector is non-zero.
Two special cases for s = t*t arise when (a) t is parallel to V and (b) t is parallel to L.(a) In the first case the previous decomposition of the translation can be done using f itself: t = f₂*f and the final product s = f₂*f*f = f₂ is the reflexion on L₂ in the direction of (t). (b) In the second case we have again an affine glide reflexion.

9. General products of reflexions

As seen by the previous examples, special cases of products of reflexions have interesting geometric properties. The study of more general products of two or more reflexions is also an interesting subject revealing several geometric properties of the affine transformations, used in many applications especially in computer science graphics questions. A general reference for the study of affinities and in particular reflexions is the book by Coxeter [CoxIntro, pp. 203]. A more advanced exposition is to be found in [VeblenYoung, II, p. 109]. A sequel to this discussion is to be found in the file Affine_Reflexion.html .

Bibliography

[CoxIntro] Coxeter, H. S. M. Introduction to Geometry, 2nd ed.. New York, John Wiley, 1972
[VeblenYoung] Oswald Veblen, John Wesley Young Projective Geometry I, II Ginn and Company, New York, 1917

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