[alogo] 1. Harmonic perspectivity

This is a special case of projectivity (see Projectivity.html , known also under the name harmonic  homology),
defined  by fixing a  point O,called center, and a line (a), not containing O, called   axis of the perspectivity. The perspectivity with this data, is defined  by the following recipe. For each point X, not  coincident with O or (a), define the intersection point HX of line [OX] and (a). Set  F(X) = Y, with Y  on line [OX], such that (O,HX,X,Y) = -1, i.e. X and Y are harmonic conjugate to O and HX. These are the main  properties of harmonic perspectivities:
[1] Point O and the points of line (a) are the only fixed points of F.
[2] F2 = 1 (is an  involutory map or involution).
[3] Every involutive projectivity is defined completely by two pairs of homologous points (X,Y)  and  (X',Y').
[4] Every involutive projectivity is a harmonic perspectivity.

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To prove [3] consider the intersection points O, O' of line-pairs (XY, X'Y') and (XX',YY') respectively. Define line (a) to be the harmonic conjugate of line OO' with respect to lines (O'X, O'Y).
The harmonic perspectivity F is completely determined by O and (a) and maps X to Y and X' to Y'. The same figure may be used to prove [4]. In fact, given the involutive projectivity G, select to points X, X' and their images Y = G(X), Y' = G(X').  Define as above the harmonic perspectivity F, mapping X to Y and X' to Y', and show that the composition F-1*G = 1 (the identity).

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The above figure has a worth noticing symmetry. In fact, one could interchange the symbols X' and Y and apply the same reasoning to create a unique perspectivity F', mapping X to X' and Y to Y' and having O' and (c) as center and axis. (c) being the harmonic conjugate of line OO' with respect to the pair of lines (OX, OX').
By the well known properties of complete quadrilaterals (see Harmonic.html ) (a) and (c) intersect at point O'', which is the intersection point of the diagonals of the quadrilateral XYY'X'.

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[alogo] 2. Related subjects

Notice that every conic (c) having the axis (a) and the center O as polar and respecctive pole,
remains invariant under F. See Fregier_Involutive.html for a discussion on that.
There is a third harmonic perspectivity F'' connected with the above figure, and having O'' and (b) as center and respective axis. The related configuration of the three perspectivities is discussed in FourPoints.html .
Harmonic perspectivities are special cases of homologies, obtained when the homology coefficient k = -1. The properties of general perspectivities are discussed in Perspectivity.html .
Using as model of the projective plane the projectification of the euclidean plane, the harmonic perspectivities with axis coinciding with the line at infinity, restricted on the euclidean plane,
coincide with the point-symmetries. See a use of this remark in the file ElationDecomposition.html .

[alogo] 3. Harmonic perspective triangles

Two triangles ABC, A'B'C' such that A'=f(A), B'=f(B), C'=f(C) are related by a harmonic perspectivity f, are called harmonic perspective triangles. They are discussed in the file PerspectivityAndPerspectiveTriangles.html .

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