If point A is

If point A

The perspectivity F with this data, is uniquely defined by the requirement to fix O (F(O)=O) and map F(A

[1] By Desargues theorem, the corresponding sides of the triangles intersect at three collinear points C

[2] Line (a) containing the three points {C

[3] The three points {C

[4] For two points X, X' and their images Y=F(X), Y'=F(X'), lines XX' and YY' intersect on the axis of F.

[5] A perspectivity is completely determined by giving its center O, its axis (a) and (X,Y), later being a couple of homologous points (Y=F(X)).

[6] Algebraically, perspectivities are characterized by the fact that they have a line (a) consisting entirely of fixed points. They are represented by matrices having two real and equal eigenvalues.

In the case of homologies, property [4] implies that the cross ratio (O,H

Homologies with

The discussion continues in the file Elation.html .

[1] If they are

[2] If D is not at infinity, then the axis is the line at infinity and the perspectivity is a

[3] If D is at infinity, then the axis is left pointwise fixed and every line in the direction determined by D is left invariant. For a point X and its image Y=F(X) it is HX/HY=k a constant, where H is the intersection point of the line XD with the axis. In particular, k=-1 gives an

[4] If they are

[5] If the axis of the elation is the line at infinity, then every line maps to a parallel line and the lines in the direction of the center O remain invariant, hence it is a

[6] If the axis of the elation is not the line at infinity, then it preserves again all lines in the direction of O, through which also passes the axis a. Thus all parallels to a are preserved, hence the map is a

PerspectivityAndPerspectiveTriangles.html

Elation.html

Harmonic_Perspectivity.html

PerspectivityThroughMatrix.html

ProjectivityResolutionPerspectivities.html

Projectivity.html

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