HarmonicPerspectivityExample

1. Harmonic Perspectivity example

Consider four points {A,B,C,D} of the projective plane in general position. Define the projectivity F by the requirements:
(i) A, B are fixed by F (F(A)=A, F(B)=B),
(ii) C, D are interchanged by F (F(C)=D and F(D)=C).
These conditions imply that F is a harmonic perspectivity with axis (a) the line AB, and center the point O, which is the harmonic conjugate of E with respect to C, D, point E being the intersection point of AB and CD.

To prove it realize first that E is fixed by F (F(E) = E). This implies that line a = AB, containing E, has three fixed points, hence consists entirely of points fixed by F. Then, considering the cross ratio (D,C, E, O), prove that O is also fixed by F. This shows that F is a perspectivity (see Perspectivity.html ), hence lines DY and CX intersect on (a) and the cross-ratio (O,H_X, X,Y) = (O,E, C,D) = -1.

2. Similar examples

Consider four points {A,B,C,D} of the projective plane in general position. Define the projectivities {F₁, F₂} by the requirements:
(i) F₁ fixes {A, C} and interchanges {B,D},
(ii) F₂ fixes {B, D} and interchanges {A,C}.
These conditions imply that {F₁, F₂} are harmonic perspectivities. The axes of {F₁,F₂} are respectively {AC, BD}. The centers of the perspectivities are correspondingly {E, H} on the diagonal GF of the corresponding complete quadrilateral of ABCD.
The composition F₃=F₂F₁ is also a harmonic perspectivity with center the intersection point O of diagonals {BD, AC} and axis the other diagonal FG of the corresponding complete quadrilateral.

1. Harmonic Perspectivity example

2. Similar examples

See Also