Here I consider affinities f (see Affinity.html ) of the plane and classify them according to the number of their fixed points i.e. points P such that f(P)=P.
In doing that I take into account also the behavior of f on the extended plane, including the points at infinity. These extra points of theplane can be identified with the common direction of parallel lines, a line L defining a point at infinity since it determines a direction of parallel lines.
As it will be seen below, affinites always posses fixed points which can be ordinary points or points at infinity.
In the following I identify the (affine) plane with the set of tripples:
The set of points at infinity is identified with tripples representing directions:
since the two tripples define the same direction in the plane. A fixed point at infinity of an affinity f corresponds to a line L which maps under f to a line L' parallel to L. For a more formal treatment on affinities look at [Audin, pp. 7].
The generic Affinity ( Affinity.html ) has a single fixed point in the plane. This can be seen by considering a parallelogram ABCD and its image A'B'C'D' under the affinity f. Then consider the intersections of corresponding opposite sides. Let us start with the side-pair (AB, CD) and its image-pair (A'B', C'D'). The intersection points {E, F} correspondingly of the pairs (CD, C'D'), (AB, A'B') define a line and consider a point P on this line. Let {E', F'} be the images of these points and P' the image of P. By the property of affinities to preserve ratios PE/PF = P'E'/P'F', hence either E'F' is parallel to EF or there is an intersection point O of lines EF and E'F'. Assuming the generic case of intersection we see easily that O is a fixed point of the affinity. In fact, PP' is always parallel to A'B', hence for Ptending to O its image-point P' tends also to O.Thus, if there is a single fixed point O, this is on line EF. A similar argument can be applied to the other pair of parallel sides (AD, BC), their images(A'D', B'C') and the corresponding intersection points {G, H} of side-pairs (AD, A'D') and (BC, B'C'). Thus, in the generic case the fixed point is the intersection point O of the two lines EF and GH ([Kovacs, p. 103]).
A special case which is possible is that line E'F' coincides with EF. In this case the whole line EF consists of fixed points of the affinity. By the preservationof ratios it must be EC/ED = E'C'/E'D' = EC'/ED'. Hence CC' and DD' are parallel and EF divides these lines in the same ratio. The figure below illustrates this case.
In that case every point I not lying on the fixed line EF maps to a point I' on the parallel to CC' from I and such that KI'/KI is constant equal to k,
independently of the position of I. Such affinities are called axial homologies (or strains ([CoxIntro, p. 203]) with axis EF and homology ratio k.
The (fixed) direction of lines II' is called conjugate direction of the homology.
In the case k = -1, the axial homology is called an affine reflection. For every point I, the segment II' has then its middle on the axis, the line II' beingalso always parallel to a fixed direction CC'.
The properties stated above can be proved easily. First prove that line BC and its image-line B'C' intersect also on the axis EF. This is seen by findingthe intersection point of BC with EF and showing that B'C' passes also through that point. Analogous is the proof of the statement about II'.
Remark Taking the coordinate axes along two special lines: a) the line EF and b) a line parallel to the direction II', the affinity is represented through the matrix:
It may happen that line E'F' is parallel to EF (point O at infinity). In this case, illustrated by the figure below, the points P of line EF map to points P' onE'F' so that EE'P'P is a parallelogram. Thus, composing the affinity f with the translation g which sends P' back to P we obtain a new affinity f'=g*f,
which leaves all points of the line EF fixed. Thus, in this case the original affinity can be written as a composition f = g-1f' i.e. as a product of anaxial homology and a translation.
By the arguments of the previous section the affinity f' is an axial homology. Thus considering the parallelogram A''B''C''D''=g(A'B'C'D'), lines{AA'', BB'', CC'', DD''} are all parallel and intersect the axis EF of f' correspondingly at points {A*, B*, C*, D*} such that the ratiosA*A/A*A'' = B*B/B*B'' = ... = D*D/D*D'' = k are all equal.
In this case the original affinity f cannot have a fixed point. In fact, if there was such a fixed point O, then the line L1 through it parallel to EE' would be invariant under f. The same would be true also for the line L2 through it and parallel to EF. Then it would follow that all lines L parallel to L1 are alsoinvariant and this would imply that all points of L2 are fixed. Obviously the same property would be valid for the inverse map and then A'B'C'D' wouldmap by f-1 onto a parallelogram with lines (CD, C'D') coinciding and also lines (AB, A'B') coinciding, hence {E,F} would not exist.
But this was a basic assumption from the beginning of the discussion. This contradiction proves the claim about the non-existence of fixed point in this case.
Remark In this case though the affinity has two fixed points at infinity, coinciding with the direction of the axis and the conjugate to it directionof the axial homology f'. Thus in case the affinity has no ordinary fixed points it has at least two fixed points at infinity.
Section-1 handles the case of an affinity with a single fixed point on the plane. Section-2 handles the case of a whole line which remains fixed (the homology axis) and a point at infinity which remains fixed (the one determined by the common direction of all segments II'). Section-3 handles the case of one single point at infinity which remains fixed. This is the point determined by the common direction of the parallels {EF, E'F'}. Combining ordinary fixed points and fixed points which lie at infiinity we get two more kinds of affinities: The first are those which leave the whole line at infinity fixed, i.e. those which map every line to a parallel line (see section-6 below).
Such affinities are called dilatations ([CoxIntro, p. 194], section-6 below). The second kind are those which have an ordinary fixed point and a second fixed point at infinity.
In the later case one can easily find two invariant lines passing through the fixed point of the affinity. The directions of these lines represent two points at infinity which are fixed by f.
In fact, assume that line L maps to a parallel line L', so that the common direction of {L, L'} represents the point at infinity fixed by the affinity f. Assume also that A is the ordinary fixed point of the affinity and draw L0 parallel to L through A. Then, by the invariance of the direction of L follows that L0 is invariant under f, hence a parallelogram ABCD with sides {AD, BC} lying respectively on {L0, L} maps to a parallelogram AB'C'D' with sides {AD', B'C'} respectively on {L0, L'}. Then lines {BB', CC'} meet at a point E or they are parallel. Assuming that the lines intersect at the ordinary point E we can show that line AE is invariant under f. In fact every point P on line L maps to a point P' on L', such that PP' passes through E. This follows by the equality of ratios PB/PC = P'B'/P'C'. In particular the intersection point Q of EA with L maps to the intersection point Q' of EA with L' and this shows the invariance of line AE. This implies that every line parallel to AE maps also to a line parallel to AE i.e. the direction of AE represents a second point at infinity which is fixed by f.
Remark-1 The fixed points at infinity demonstrate a different behaviour from that of ordinary points, regarding the line they define (the line at infinity). In fact two ordinary fixed points define a line, which remains pointwise fixed (preservation of ratios). In contrast, as is here the case, two fixed points at infinity do not imply that the whole line at infinity remains fixed. Also three ordinary fixed points for f imply that later coincides with the identity. In contrast, as is here the case, two fixed points at infinity plus one ordinary do not imply that f is the identity. Also two ordinary fixed points plus one point at infinity (define an axial homology, as in section-2) do not imply that f coincides with the identity.
Remark-2 Introducing coordinates along the axes AD, AQ we can represent f through a matrix M of the above form. Its first representation as the product on the right shows that such a map is the composition of two axial homologies (section-2). In fact, the simple matrices on the right represent obviously axial homologies. As seen from the matrix-representation or the figure, the product is commutative i.e. the factors taken in either order represent by their products the same affinity. The two factors representing f as product are two linked strains in the sense that the axis of one defines the conjugate direction of the other. The second representation as a product shows that the affinity can also be expressed as the product of a strain and a homothety.
Remark-3 The same behavior occurs when lines BB' and CC' are parallel i.e. E goes to infinity. Then line AE becomes parallel to these two lines and, as in the previous case, it remains invariant under the affinity f.
In the case the affinity has two ordinary fixed points {A, B}, it is completely determined by a third point C and its image C'. In this case all points of line AB remainfixed and in addition line CC' remains invariant under the affinity. This implies that all lines parallel to CC' map also to a parallel line. The case is identical with the oneof section-2. Considering lines {IC, I'C'} for an arbitrary point I and its image I' we show again easily that the ratio KI'/KI is constant and independent of the positionof I.
If the affinity has the line AB fixed and maps a point C to C' so that CC' is parallel to AB, then it is called a shear ([CoxIntro, p. 203]). In that case lineCC' remains invariant under f and it follows easily that all lines parallel to AB remain also invariant under the affinity. Besides, it is easily seen that on each lineparallel to AB the affinity coincides with a translation. In the figure below this is visible from the fact that CC' and II' are segments of equal length.
Taking the coordinate axes along lines AB and AC one can easily see that the affinity can be represented by a simple matrix of the form:
Obviously shears have no other fixed points outside the line AB, which is called their axis. Besides for every pair of points (C, P) their images (C', P') define lines CPand C'P' intersecting on their axis.
Remark Affinities having a line L consisting entirely of fixed points are called axial affinities. By the results of this section and section-2 such affinities are either strains or shears. The difference of the two kinds is the existence of an additional fixed point at infinity, which for strains does not lie on the (extended) L, whereas for shears it does lie on (the extended) L.
In the case the affinity has three points at infinity fixed, then it fixes every point at infinity and is a dilatation (section-4). Next figure illustrates why. TriangleABC has its sides parallel to the three directions determined by the three distinguished points at infinity. By assumption its image-triangle A'B'C' will have its sidesparallel to corresponding sides of ABC, hence it will be (anti) homothetic to it (or a translate of it).
It follows that the original affinity coincides with the (anti) homothety or translation associating {A,B,C} to {A',B',C'}. These are maps fixing every point of theline at infinity, hence the proof of the claim.
Remark-1 It follows that translations are characterized by fixing every point at infinity and having no other (ordinary) fixed points. Equivalently, theyare characterized by leaving three points at infinity invariant and having no other (ordinary) fixed point.
Homotheties in turn are characterized by having a single ordinary fixed point and three (hence all) fixed points at infinity.
Remark-2 The set of all dilatations is a Group. Thus, composing dilatations or taking inverses we obtain again dilatations. The subset of translationsis a subgroup of this group. The composition of a homothety and a translation is in general a homothety. The composition of two homotheties can be a homothety, but in some cases (see Homotheties_Composition.html ) can be also a translation. Thus, the set of homotheties, being not closed under composition, isnot a group.
Assume that the affinity has no fixed points at all (neither ordinary nor at infinity). Then the cases of sections 1,2 and 3 must be excluded, since in these cases theyappear fixed points. This means that taking a parallelogram ABCD and its image A'B'C'D', the line-pairs (AB, A'B') and (CD, C'D') must be parallel, otherwise we would obtain points {E, F} as in section-1 and fall back to those cases. But this shows that AB and CD define two directions or points at infinity which are fixedby the affinity. A contradiction proving that every affinity has fixed points either ordinary or at infinity.
An affinity having two fixed points at infinity defines two directions of lines which map to parallel lines. If in addition there is an ordinary fixed point, then the caseis the one described in section-4 by the product of two strains. If there is no fixed ordinary point, then taking any point O and two lines through it {L,L'} parallelto the fixed directions and applying f to this system we obtain the point O'=f(O) and lines {N=f(L), N'=f(L')} correspondingly parallel to {L,L'}. Let g be the translation sending O' back to O. Then the composite transformation f' = g*f fixes O and leaves invariant lines {L, L'}, hence, by section-4, it is the product oftwo strains s2*s1. Thus the original affinity is the product of three special affinities f = g-1*s2*s1.
If the affinity has only one fixed point at infinity then there is a unique direction of lines L such that L'=f(L) is parallel to L. If the affinity has in addition an isolatedordinary fixed point, then the results of section-4 can be applied and show that the affinity is again the product of two strains and there is also one additional fixedpoint at infinity. This contradicts our assumption, hence there is no affinity with exactly one ordinary fixed point and a fixed point at infinity.
If the affinity has exactly one fixed point at infinity and no other fixed points, then take a point O and a line L through it in the direction of the fixed point. Applyingf we find O'=f(O) and L'=f(L) parallel to L. Let then g be again the translation sending O' back to O. The composite affinity f'=g*f leaves O fixed and line L invariant,
hence by section-4 it determines an additional fixed point at infinity, which is also fixed point of the original f. This contradicts again the assumption on f, hence there is no affinity with exactly one fixed point at infinity and no other fixed points.
1) Every affinity has a fixed point, which is an ordinary point or a point at infinity. This means that every affinity either has an ordinary fixed point or there is a line L which maps by the affinity to a parallel line L'.
2) An affinity with three fixed ordinary points is the identity.
3) An affinity with two fixed ordinary points {A,B} has the whole line AB consisting of fixed points and it is a strain or shear (section-4, -5), hence defines also
two/one additional fixed points at infinity.
4) An affinity with a single ordinary fixed point, either has no other fixed points at infinity (section-1) or has two fixed points at infinity (section-4) or is a homothety
and leaves every point at infiinity fixed (section-6, dilatations).
5) An affinity with no ordinary fixed point and three fixed points at infinity (hence all points at infinity fixed) is a translation.
6) An affinity with no ordinary fixed point and one fixed point at infinity has a second point at infinity and it is a product of two strains and a translation (section-4).
7) An affinity with no ordinary fixed point and two fixed points at infinity is a product of two strains and a translation (section-4).
There are numerous results concerning the representation of affinities as products of other simpler ones, or characterizing the kind of an affinity by its behavior onsome points or some lines. The following are some characteristic examples.
1) If, for a given affinity, every noninvariant point lies on at least one invariant line, then the affinity is either a dilatation or a shear or a strain ([CoxIntro, p. 203]).
2) Any translation or half-turn (anti-homothety with ratio = -1) or shear can be expressed as the product of two affine reflections ([CoxIntro, p. 208]).
3) Any affine transformation can be obtained as a product of a similarity (see Similarity.html ) and an axial affine transformation (shear of strain, [Nemet, p. 232]).
I leave the first two as exercises and prove the last statement.
Let the affinity f be defined by the vertices of triangle ABC and their images which are the vertices of the triangle A*B*C*. Then consider a similarity s mapping
side AB to A*B* and triangle ABC to triangle A*B*C', where C' = s(C). In addition consider the strain (or shear) t fixing {A*, B*} and mapping C' to C*.
Obviously f = t*s (composition), thereby proving the claim.
See Also
Affinity.html
Homotheties_Composition.html
Similarity.html
Bibliography
[Audin] Audin, Michele Geometry Berlin, Springer, 2002
[CoxIntro] Coxeter, H. S. M. Introduction to Geometry, 2nd ed.. New York, John Wiley, 1972
[Kovacs] Zoltan Kovacs On the fixed points of an affine transformation Teaching Mathematics and Computer Science 2006(IV)
[Nemet] Istvan Krisztin Nemet Remarks on the concepts of affine transformation... Annales Mathematicae et Informaticae 32 (2005)
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