[alogo] 1. Elation definition

These are projectivities characterized by the existence of a line (a) consisting entirely of fixed points, no other fixed point existing outside this line, and a special point A on this line such that all lines through A remain invariant under the map. Line (a) is called the axis and point A the center of the elation.
It can be easily shown that the requirement, to have a point A such that all lines through it remain invariant is a redundant one. It is implied by the property of the axis to include all the fixed points of the map and not having other fixed points outside.

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In fact, consider two points {D,E} and their images {D'=F(D), E'=F(E)} under the elation. Lines {DD',EE'} are invariant under F and must intersect at a point A on the axis (a) of the elation. If this were not the case and A were outside (a), then A, being the intersection point of two invariant lines would be fixed and outside (a) contradicting the assumption on F.

[alogo] 2. Harmonic homologies composition

[1] The product FC = FBFA (composition of transformations) of two harmonic perspectivities FA, FB with the same axis (e) and different centers A, B is an elation (FC) with axis (e) and center the intersection point C of (e) and the line AB.
[2] The product F' = FAFB of the harmonic perspectivities of the previous perspectivities FA, FB in reverse order is the inverse of FC. It is also an elation with the same center C and axis (e) but different from FC. For every point X, the images Z=FC(X), Z*=F'(X) are harmonic conjugate with respect to {X,C}.
[3] The map Z --> Z* is the commutator F'(FC)-1 = (FAFB)2 = F'2, which is also an elation with the same center and axis as FC.

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Take Y=FA(X), Z=FB(Y) for an arbitrary point X. Points {X,Z,C} are collinear since (X,Y,A,X')=(Y,Z,B,Y')=-1. This proves [1]. Point X*=FB(X) is on line CY because of the harmonic relation. Line X*Z intersects (e) at W, which is the intersection of lines {AX, e} because of the harmonic bundle at C. For the same reason (X*,Z,W,W')=-1, by projecting from X => (B,C,A,W')=-1 and by projecting this from X* => (C,X,Z,Z*)=-1. [3] follows from the preceding.

[alogo] 3. Elation composition

[1] The product G of an elation FE, with center E, and a harmonic perspectivity FC, with center C, the two maps sharing the same axis (a), is a harmonic perspectivity FD. The center D of FD lying on line EC.
[2] Every elation FE is the product of two harmonic perspectivities FC*FD, the two perspectivities sharing with FE the same axis. This decomposition can be done in infinite many ways by selecting a point C arbitrarily, outside the axis (a) of the elation.
[3] The product of three harmonic perspectivities sharing the same axis (a) and having their centers {A,B,C} aligned is a harmonic perspectivity.

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Line CE is invariant under the composition G = FCFE. Point E is a fixed point of G. Hence G restricted on CE is represented by a Moebius transformation (ax)/(cx+d), taking x=0 to represent E. Then the other fixed point will satisfy cx+(d-a)=0, and since (d-a)2 is the determinant of the Moebius transformation in this case, there will be a second fixed point at x=(a-d)/c. Thus the composition G will fix the line (a) and the point D on CE corresponding to this value for x. Thus G will be a homology. It is then easily seen that the homology coefficient will be -1 by considering the cross ratio (C,Z',Z,V)=-1 , which by the definition of harmonic perspectivity at C, implies through radial projection from E that (V',D,X,V)=-1. Hence the map is a harmonic perspectivity. This proves [1]. [2,3] are consequences of [1].

[alogo] 4. Remarks

Remark-1 Previous figure shows the representation of the elation as a product of two harmonic perspectivities {FA, FB} with common axis. Y=FA(X), Z=FB(X), V = FC(X).
Remark-2 The figure below shows the application F(C) = D defined by a line (a) and two points {A,B} not on it. It is given by constructing first F as intersection of (a) and BC and then taking the intersection D of lines {EC, AF}. This map is identical with the elation with axis (a), center E and mapping B to A. This elation however is not one of {FBFA, FAFB}.

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Remark-3 An elation with axis (a) and center E, as above, cannot have a second fixed point E' different from E along AB. This would contradict its definition. Consequently the line-projectivity (f) induced on the invariant lines passing through E cannot be an involution. This would imply the existence of a second fixed point on line AB.
Remark-4 By selecting a projective basis in the line AB we can represent (f) of the previous remark through a Moebius transformation. This, according to remark-3 will have only one fixed point (E) on line BC, thus it is a parabolic Moebius transform.

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