Projectivity

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Projectivity: Definition

If \(L \) and \(L'\) are two ranges (lines) in a projective plane \(\pi\), then a projectivity \(L \) to \(L'\) is composition of finite number of perspectivities from \(L \) to \(L'\).
We denote the projectivity \( f:L\to L' \) by \( f:L \sim L'\) .
If f is a projectivity from \(L \) to \(L'\) then
either f is itself a perspectivity
or there exists finite number of perspectivities such that
\( f: L \overset{u_1}{\pi} L_1 \overset{u_2}{\pi} L_2\overset{u_3}{\pi} L_3 \ldots \overset{u_n}{\pi} L_n \overset{u_{n+1}}{\pi} L'\)

In this composition, there are (n+1) number of compositions from \(L \) to \(L'\). Therefore, \( f:L\to L' \) is a projectivity.

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If \(a,b,c\in L\) and \(a' ,b' ,c' \in L'\) , then there exist a projectivity \(f:L (a,b,c)\sim L' (a' ,b' ,c')\)

Proof

Given, \( a,b,c\in L\) and \( a' ,b' ,c' \in L'\).

By P1, there exist a line cc' .

By P2, the lines aa' , bb' , ab' (M) must meets cc' , say respectively at r,s,t.

Now
There exist a perspectivity \( f_1: L \to M\) given as
\( L(abc) \buildrel s \over \pi M(ab' t) \)

Also
There exist a perspectivity \( f_2: M \to L'\) given as
\( M(ab' t) \buildrel r \over \pi L' (a' b' c' ) \)

Hence
L(a,b,c) and L'(a' ,b' ,c') are related by of two perspectivities given as
\( L(abc) \buildrel s \over \pi M(ab' t) \buildrel r \over \pi L' (a' b' c' ) \)
Thus,
\( f:L (a,b,c)\sim L' (a' ,b' ,c')\) is a projectivity
This completes the proof.

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Let \(a,b\in L\) and \(a' ,b' \in\ L' \). If \(f:L \sim L'\) is a projectivity, then there exists a 2x2 non-singular matrix M such that \( (\lambda' ,\mu')=(\lambda,\mu)M\).

Proof
Given \( f:L \sim L'\) is a projectivity, then
There exist a finite number of perspectivities from \(L\) to \(L'\), say, it is given by
\( f: L \overset{u_1}{\pi} L_1 \overset{u_2}{\pi} L_2\overset{u_3}{\pi} L_3 \ldots \overset{u_n}{\pi} L_n \overset{u_{n+1}}{\pi} L'\)
Here, \(L_n \overset{u_{n+1}}{\pi} L'\) is a perspectivity, thus
\( (\lambda',\mu')=(\lambda_n,\mu_n)M_{n+1} \) (A)
Also, \(L_{n-1} \overset{u_n}{\pi} L_n \) is a perspectivity, thus
\( (\lambda_n,\mu_n)= (\lambda_{n-1},\mu_{n-1}) M_n \) (i)
Using (i), the equation (A) becomsas
\( (\lambda',\mu')= \) \( (\lambda_n,\mu_n) \) \(M_{n+1}\)
or \( (\lambda',\mu')\)= \((\lambda_{n-1},\mu_{n-1}) M_n \) \(M_{n+1} \) (A)
Continuing this process, we arrive at a situation
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\( (\lambda',\mu')=(\lambda_2,\mu_2) M_3...M_nM_{n+1} \) (A)
Now,
\(L_1 \overset{u_2}{\pi} L_2 \) is a perspectivity, thus
\( (\lambda_2,\mu_2)= (\lambda_1,\mu_1) M_2 \) (ii)
Using (ii), the equation (A) becomsas
\( (\lambda',\mu')= \) \( (\lambda_2,\mu_2) \) \(M_3...M_nM_{n+1} \)
or \( (\lambda',\mu')\)= \((\lambda_1,\mu_1) M_2 \) \(M_3...M_nM_{n+1} \) (A)
Finally,
\(L \overset{u_1}{\pi} L_1 \) is a perspectivity, thus
\( (\lambda_1,\mu_1)= (\lambda,\mu) M_1 \) (iii)
Using (iii), the equation (A) becomsas
\( (\lambda',\mu')= \) \( (\lambda_1,\mu_1) \) \(M_2M_3...M_nM_{n+1} \)
or \( (\lambda',\mu')\)= \((\lambda,\mu) M_1 \) \(M_2M_3...M_nM_{n+1} \)
or \( (\lambda',\mu')=(\lambda,\mu)\) \( M_1 M_2M_3...M_nM_{n+1} \)
or \( (\lambda',\mu')=(\lambda,\mu)\) \( M \)
where \( M=M_1 M_2M_3...M_nM_{n+1} \) is a 2x2 non-singular matrix.
Thus, for a projectivity \(f:L \sim L'\), there exists a 2x2 non-singular matrix M such that
\( (\lambda' ,\mu')=(\lambda,\mu)M\)

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Projectivity from \( L \to\ L'\) is a bijection and its inverse is also projectivity from \( L' \to\ L\).

Proof
Let \( f:L\to L'\) is a projectivity, then
There exist a finite number of perspectivities from \( L \to\ L'\), say, it is given by
\( f: L \overset{u_1}{\pi} L_1 \overset{u_2}{\pi} L_2\overset{u_3}{\pi} L_3 \ldots \overset{u_n}{\pi} L_n \overset{u_{n+1}}{\pi} L'\)
Let say, these perspectivities are expressed as
\( f=f_1 \circ f_2 \circ \ldots \circ f_n \) where \( f_1,f_2,\ldots,f_n\) are perspectivities
Since, \(f_i;i=1,2,…,n\) is bijection, f is bijection.
Next,
\(f^{-1}=f_n^{-1} \circ f_{n-1}^{-1} \circ \ldots \circ f_1^{-1}\) where \( f_n^{-1},\ldots,f_1^{-1}\) are perspectivities
Here,
\( f^{-1}\) is composition of finite number of perspectivities
Therefore, \( f^{-1}\) is a projectivity from \( L' \to\ L\)
This completes the proof.