Cross ratio. Harmonic conjugates. Perspectivity. Projectivity

Definition Let \(A\), \(B\), \(C\), and \(D\) be colinear points. The cross ratio of the pairs of points \((A,B)\) and \((C,D)\) is \[ \mathcal{R}(A,B;C,D) = \frac{\overrightarrow{AC}} {\overrightarrow{CB}}:\frac{\overrightarrow{AD}} {\overrightarrow{DB}}. \quad\quad\quad\quad\quad (1) \]

Let \(a\), \(b\), \(c\), \(d\) be four concurrent lines. For the given lines \(p_1\) and \(p_2\) let us denote \(A_i=a\cap p_i\), \(B_i=b\cap p_i\), \(C_i=c\cap p_i\), \(D_i=d\cap p_i\), for \(i=1\), \(2\). Then \[\mathcal R(A_1,B_1;C_1,D_1)= \mathcal R(A_2,B_2;C_2,D_2).\quad\quad\quad\quad\quad (2)\]

Thus it is meaningful to define the cross ratio of the pairs of concurrent lines as \[ \mathcal R(a,b;c,d)=\mathcal R(A_1,B_1;C_1,D_1). \quad\quad\quad\quad\quad (3)\]

Assume that points \(O_1\), \(O_2\), \(A\), \(B\), \(C\), \(D\) belong to a circle. Then \[ \mathcal R(O_1A,O_1B;O_1C,O_1D) = \mathcal R(O_2A,O_2B;O_2C,O_2D).\quad\quad\quad\quad\quad (4)\] Hence it is meaningful to define the cross-ratio for cocyclic points as \[ \mathcal R(A,B;C,D) =\mathcal R(O_1A,O_1B;O_1C,O_1D).\quad\quad\quad\quad\quad (5)\] Assume that the points \(A\), \(B\), \(C\), \(D\) are colinear or cocyclic. Let an inversion with center \(O\) maps \(A\), \(B\), \(C\), \(D\) into \(A^*\), \(B^*\), \(C^*\), \(D^*\). Then \[\mathcal R(A,B;C,D)=\mathcal R(A^*,B^*;C^*,D^*). \quad\quad\quad\quad\quad (6)\]

Definition Assume that \(A\), \(B\), \(C\), and \(D\) are cocyclic or colinear points. Pairs of points \((A,B)\) and \((C,D)\) are harmonic conjugates if \(\mathcal R(A,B;C,D)=-1\). We also write \(\mathcal H(A,B;C,D)\) when we want to say that \((A,B)\) and \((C,D)\) are harmonic conjugates to each other.

Definition Let each of \(l_1\) and \(l_2\) be either line or circle. Perspectivity with respect to the point \(S\) (denote as \(\frac{S}{\overline\wedge}\)), is the mapping of \(l_1\rightarrow l_2\), such that

(i) If either \(l_1\) or \(l_2\) is a circle than it contains \(S\);

(ii) every point \(A_1\in l_1\) is mapped to the point \(A_2=OA_1\cap l_2\).

According to the previous statements perspectivity preserves the cross ratio and hence the harmonic conjugates.

Definition Let each of \(l_1\) and \(l_2\) be either line or circle. Projectivity is any mapping from \(l_1\) to \(l_2\) that can be represented as a finite composition of perspectivities.

Theorem 1 Assume that the points \(A\), \(B\), \(C\), \(D_1\), and \(D_2\) are either colinear or cocyclic. If the equation \(\mathcal R(A,B;C,D_1)= \mathcal R(A,B;C,D_2)\) is satisfied, then \(D_1=D_2\). In other words, a projectivity with three fixed points is the identity.

Theorem 2 If the points \(A\), \(B\), \(C\), \(D\) are mutually discjoint and \(\mathcal R(A,B;C,D)=\mathcal R(B,A;C,D)\) then \(\mathcal H(A,B;C,D)\).