Affine_Reflexion

1. A lemma on two reflexions

The basics on reflexions are discussed in the file Affine_Reflexion_Basics.html . Here the discussion continues with the investigation of products of reflexions. A crucial role in the
discussion plays the following fact: Given a triangle ABC and point D, let EFG be the triangle of side-middles of ABC. Extend FG to hit DE at L and take on it LH=LG. Draw
parallel HI to AC from H having its middle J on DF. The so created triangle GHI has the middle K of the side GI on line DA.

The lemma is proved, together with some other related properties of the figure, in section-2 of the file Line_Locus_Criterium.html . From the viewpoint of affine reflexions it implies a
certain propetry on the composition of two reflections examined in the next section.

2. Two reflexions and a line

I denote by f = (L,V) the affine reflexion whose axis of fixed points is the line L and the conjugate direction is given by the direction of line V. These data completely determine
the transformation f, the only shortcoming of the notation being the ambiguity in V, which can be replaced by any parallel to it line. The following property holds for two reflexions
f_i = (L_i, V_i), i=1,2, whose axes intersect at point O.
For every line L through O there are defined three other lines {V,L₃,V₃} such that the corresponding reflexions f₃=(L₃,V₃) and f=(L,V) have composition the identity:
f * f₃ * f₂ * f₁ = Id. (*)

The proof follows from the lemma of the previous section. In fact, for every point A on line L, taking B = f₁(A) and C = f₂(B) defines a triangle which remains similar to itself
as A varies on L. Thus, the middles F of the sides AC of these triangles vary on a line L₃, while the sides themselves remain parallel to a fixed direction represented by a line V₃.
By the aforementioned lemma, taking the middle G of AB and defining H = f₂(G), I = f₃(H), where f₃ = (L₃,V₃), we obtain triangle GHI whose side IG has its middle on L.
As A varies on L the direction V of GI remains constant and this defines the reflexion f = (L,V).
The composition represented by the left side of (*) is the identity for every point A on L. This is equivalent to the existence of triangle ABC which remains similar to itself for all
positions A on L. The same composition is also constant for every point G on line L₁. This is again equivalent to the existence of triangle GHI which also remains similar to itself
for all positions of G along L₁. Thus the two affine transforms represented by the two sides of (*) coincide along lines L and L₁, hence they coincide everywhere (see Affinity.html ).

Remark-1 The above property is equivalent to the fact that taking any point X on the plane and defining X'=f₁(X), X''=f₂(X'), X'''=f₃(X'') we have f(X''') = X. The
quadrilaterals with vertices {X,X',X'',X'''} have their sides parallel to the V_i' s and the middles of corresponding sides lying on lines L_i.
Remark-2 The property can be also interpreted as a property of the composition f₂*f₁ of two reflexions. In fact, it says that for any given line L through the intersection
point O of the axes of {f₁, f₂} there are lines {V, L₃, V₃} defining two other reflexions f₃ = (L₃,V₃) and f=(L,V) such that
f₂ * f₁ = f₃ * f.
This can be also translated in the following way: The product of two reflexions f₂*f₁ can be expressed as product of two other reflexions f₃*f of which the axis of f can be
an arbitrary line L through the intersection point of the axes of f₁, f₂.

3. Triangular constellations of reflexions

The figure of the previous section can be interpreted in another way. The basic triangle ABC and the point O can be considered to define the basic configuration of three reflexions
whose axes pass through the midpoints of the sides of the triangle and also through the point O. I call such configurations triangular constellations of reflexions with center at O.

The three reflexions are f₁ = (OG, AB), f₂ = (OE, BC), f₃ = (OF, CA). There is then defined a fourth reflexion through f = (OA, IG) and last equation of the previous section
can be written in the form
f₃ * f₂ * f₁ = f,
showing that the product of the three reflexions of a triangular constellation is a reflexion with axis also passing through the center O of the constellation.

4. Non constellar concurrence

I use this term to denote three reflexions f_i = (L_i, V_i) whose axes L_i are concurrent at a point O but they do not build a triangular constellation. This means that for each
point X the successive images X' = f₁(X), X'' = f₂(X'), X''' = f₃(X'') do not close to a system of vertices of a triangle i.e. X''' is different from X.

In this case, using the results of section-2 we can reduce the product to another which is more convenient to describe geometrically.

In fact there are then two other reflexions {f' , f''} of which the first has the same axis L'=L₁ with f₁, such that f₃ * f₂ = f'' * f' and consequently
f₃ * f₂ * f₁ = f'' * f' * f₁,
whereby the composition s = (f' * f₁) of two reflexions with the same axis L₁ is a shear (see section-4 of Affine_Reflexion_Basics.html ).
This case of composition h = f * s, of a shear and a reflexion f is analyzed in the next section.

5. Shear and reflexion composition

Here I am interested to understand the composition h = f * s of a shear s and an affine reflexion f. I use a cordinate system adapted to the shear with origin at the intersection
point O of the shear and the reflexion (this conditions results from the basic assumption of the concurrence of the axes of three reflexions). The matrix S representing the shear s
has the form

and the reflexion f is represented by a matrix depending on the two lines {L, V} representing f = (L, V) (see section-2). Assume that the lines are represented by the equations
L : ax + by = 0,
V : cx + dy = 0.
If A denotes the matrix of the affine reflexion in the selected coordinate system, then the points of these lines will correspond to eigen-vectors of the matrix. Thus,
for the typical vectors (-b, a) and (-d, c), which are points respectively of L, V there will result a matrix equation

Without loss of generality we may assume that the determinant ad-bc = 1. The matrix A is then easily computed and seen to be equal to:

From this follows that the corresponding product matrix H = AS, which represents in the selected coordinate system the composition h = f *s, has the form:

The condition on the coefficients of H to guarantee the existence of two real eigenvalues is (elementary linear algebra)
trace(H)² -4det(H) > 0.
It is easily seen that det(H) = -1 and from this that the above condition becoms (2kac)² + 4 >0. Thus there are two real eigenvalues k₁, k₂ and in a basis consisting from
eigenvectors of H the matrix of H (taking into account that det(H) = -1) will take the form

These transformations interchange the points of the hyperbola xy = 1 with those of its conjugate xy = -1. Summarizing the results on three concurring reflexions (i.e.
reflexions whose axes of fixed points concurre at some point O) we have the result:
Three concurring reflexions define a positive number k and two lines through O on which their composition acts by multiplying vectors by k, respectively -1/k.
Obviously the case of three constellar reflexions corresponds to k = 1, which identifies h as the reflexion with axis coinciding with the x-axis and conjugate axis coinciding
with the y-axis.

6. Product of three reflexions

The product of three arbitrary reflexions in general position, i.e. reflexions whose axes as well as the conjugate directions are lines in general position can be reduced to cases
studied above. In fact for three such reflexions f₁=(EF,AB), f₂=(FG,BC), f₃=(GE,CD) we can rewrite the composition
                                                                                             f = f₃ * f₂ * f₁
by inserting an extra reflexion twice (since g² = Id, is the identity):
                                                                                             f = f₃ * g * g  * f₂ * f₁,
where g=(L,CD) is the reflexion with axis parallel to the one of f₃ through F, later being the intersection of the two axes of f₁ and f₂, the conjugate direction of g being also that of
f₃. By the discussion in section-5 the first part (g*f₂*f₁) of this product has a definite simple representation by a diagonal matrix. By the discussion in section-3 of the
Affine_Reflexion_Basics.html the composition of the two reflexions f₃*g is a translation.

By adopting the coordinate system in which (g*f₂*f₁) is represented by a diagonal matrix (like H' in previous section) we see that the transformation, in this case, can be
represented in the form

In this the column vectors are identified with points of the plane and (u,v) represents the translation part of the transformation corresponding to the factor (f₃*g). The
diagonal matrix in the middle represents an affine transformation (equiaffinity) which has the special name ([CoxIntro, p. 209]) hyperbolic rotation or/and Procrustean stretch. It is a
transformation preserving both the hyperbola xy=1 and also its conjugate xy = -1. Some applications of this analysis may be found in the file InscribedTria_In_Tria.html .

7. Procrustean stretch

The diagonal matrix entering the equation of the previous section can be further analyzed in a product of two simple matrices representing correspondingly two reflexions and
this in an infinity of ways.

The typical matrix representing the factors has the eigenvalues {+1, -1} and the corresponding eigenvectors {(1, a), (-1, a)}. The first represents the direction of the axis of the
reflexion and the second represents the conjugate direction of the affine reflexion.

Bibliography

[CoxIntro] Coxeter, H. S. M. Introduction to Geometry, 2nd ed.. New York, John Wiley, 1972

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