Cross Ratio Geometry
Advances for Four Co-Linear Points in the
Desargues Affine Plane-Skew Field

Let $\mathcal{P}$ be a nonempty space, $\mathcal{L}$ a nonempty subset of $\mathcal{P}$. The elements $p$ of $\mathcal{P}$ are points and an element $\ell$ of $\mathcal{L}$ is a line.

Desargues’ Axiom, circa 1630 [11, §3.9, pp. 60-61] [17]. Let $A,B,C,A^{\prime},B^{\prime},C^{\prime}\in\mathcal{P}$ and let pairwise distinct lines $\ell^{AA_{1}},\ell^{BB^{\prime}},\ell^{CC^{\prime}},\ell^{AC},\ell^{A^{\prime}C^{\prime}}\in\mathcal{L}$ such that

Then $\boldsymbol{\ell^{AC}\parallel\ell^{A^{\prime}C^{\prime}}}$.   $\blacksquare$

Addition of points in a line of affine plane: In an Desargues affine plane $\mathcal{A_{D}}=(\mathcal{P},\mathcal{L},\mathcal{I})$ we fix two different points $O,I\in\mathcal{P},$ which, according to Axiom 1, determine a line $\ell^{OI}\in\mathcal{L}.$ Let $A$ and $B$ be two arbitrary points of a line $\ell^{OI}$. In plane $\mathcal{A_{D}}$ we choose a point $B_{1}$ not incident with $\ell^{OI}$: $B_{1}\notin\ell^{OI}$ (we call the auxiliary point). Construct line $\ell_{OI}^{B_{1}},$ which is only according to the Axiom 2. Then construct line $\ell_{OB_{1}}^{A},$ which also is the only according to the Axiom 2. Marking their intersection $P_{1}=\ell_{OI}^{B_{1}}\cap\ell_{OB_{1}}^{A}.$ Finally construct line $\ell_{BB_{1}}^{P_{1}}.$ For as much as $\ell^{BB_{1}}$ cuts the line $\ell^{OI}$ in point $B$, then this line, parallel with $\ell^{BB_{1}}$, cuts the line $\ell^{OI}$ in a single point $C$, this point we called the addition of points $A$ with point $B$ (Figure 2 (a)).

Multiplication of points in a line in affine plane. Choose in the plane $\mathcal{A_{D}}$ one point $B_{1}$ not incident with lines $\ell^{OI},$ and construct the line $\ell^{IB_{1}}$. Construct the line $\ell_{IB_{1}}^{A},$ which is the only accoding to the Axiom 2 and cutting the line $\ell^{OB_{1}}$. Marking their intersection with $P_{1}=\ell_{IB_{1}}^{A}\cap OB_{1}.$ Finally, construct the line $\ell_{BB_{1}}^{P_{1}}.$ Since $\ell^{BB_{1}}$ cuts the line $\ell^{OI}$ in a single point $B$, then this line, parallel with $\ell^{BB_{1}}$, cuts the line $\ell^{OI}$ in one single point $C$, this point we called the multiplication of points $A$ with point $B$ (Figure 2 (b)).

The process of construct the points $C$ for adition and multiplication of points in $\ell^{OI}-$line in affine plane, is presented in the tow algorithm form

Addition Algorithm

Step.1:

$B_{1}\notin\ell^{OI}$

Step.2:

$\ell_{OI}^{B_{1}}\cap\ell_{OB_{1}}^{A}=P_{1}$

Step.3:

$\ell_{BB_{1}}^{P_{1}}\cap\ell^{OI}=C(=A+B)$

Multiplication Algorithm

Step.1:

$B_{1}\notin\ell^{OI}$

Step.2:

$\ell_{IB_{1}}^{A}\cap\ell^{OB_{1}}=P_{1}$

Step.3:

$\ell_{BB_{1}}^{P_{1}}\cap\ell^{OI}=C(=A\cdot B)$

In [18] and [6], we have prove that $(\ell^{OI},+,\cdot)$ is a skew field in Desargues affine plane, and is field (commutative skew field) in the Papus affine plane.

I n this section $K$ will denote a skew field [8] and $z[K]$ its center, where is the set $K$ such that

Let now $p\in K$ be a fixed element of the skew field $K$. We will denote $z_{K}(p)$ the centralizer in $K$ of the element $p$, where is the set,

$z_{K}(p)$ is sub skew field of K, but, in general, it is not commutative.

Let $K$ be a skew field, $p\in K$, and let us denote by $[p_{K}]$ the conjugacy class of $p$:

If, $p\in z[K]$, for all $q\in K$ we have that $q^{-1}pq=p.$

In the paper [23], we have done a detailed study, related to the ratio of two and three points in a line of Desargues affine plane. Below we are listing some of the results for ratio of two and three points.

For a ’ratio-point’ $R\in\ell^{OI}$, and for point $B\neq O$ in line $\ell^{OI}$, is a unique defined point, $A\in\ell^{OI}$, such that $R=B^{-1}A=r(A:B)$.

Some results for Ratio of 2-points in Desargues affine plane (see [23]).

• •

  If have two different points $A,B\in\ell^{OI}-$line, and $B\neq O$, in Desargues affine plane, then, $r^{-1}(A:B)=r(B:A)$.

• •

  For three collinear point $A,B,C$ and $C\neq O$, in $\ell^{OI}-$line, have,

  $$r(A+B:C)=r(A:C)+r(B:C).$$

• •

  For three collinear point $A,B,C$ and $C\neq O$, in $\ell^{OI}-$line, have,

  1. (1)

     $r(A\cdot B:C)=r(A:C)\cdot B.$

  2. (2)

     $r(A:B\cdot C)=C^{-1}r(A:C).$

• •

  Let’s have the points $A,B\in\ell^{OI}-$line where $B\neq O$. Then have that,

  $$r(A:B)=r(B:A)\Leftrightarrow A=B.$$

• •

  This ratio-map, $r_{B}:\ell^{OI}\to\ell^{OI}$ is a bijection in $\ell^{OI}-$line in Desargues affine plane.

• •

  The ratio-maps-set $\mathcal{R}_{2}=\{r_{B}(X)|\forall X\in\ell^{OI}\}$, for a fixed point $B$ in $\ell^{OI}-$line, forms a skew-field with ’addition and multiplication’ of points. This, skew field $(\mathcal{R}_{2},+,\cdot)$ is sub-skew field of the skew field $(\ell^{OI},+,\cdot)$.

Ratio of three points in a line on Desargues affine plane. (see [23])

Some Results for Ratio of 3-points in Desargues affine plane ([23]).

• •

  For 3-points $A,B,C$ in a line $\ell^{OI}$ of Desargues affine plane, we have that,

  $$r(-A,-B;-C)=r(A,B;C).$$

• •

  For 3-points $A,B,C$ in a line $\ell^{OI}$ in the Desargues affine plane, have

  $$r^{-1}(A,B;C)=r(B,A;C).$$

• •

  If $A,B,C$, are three different points, and different from point $O$, in a line $\ell^{OI}$ on Desargues affine plane, then

  $$r(A^{-1},B^{-1};C^{-1})=B[r(A,B;C)]A^{-1}.$$

• •

  In the Pappus affine plane, for three point different from point $O$, in $\ell^{OI}-$line, we have $r(A^{-1},B^{-1};C^{-1})=r(A,B;C)\cdot r(B,A;O).$

• •

  This ratio-map, $r_{BC}:\ell^{OI}\to\ell^{OI}$ is a bijection in $\ell^{OI}-$line in Desargues affine plane.

• •

  The ratio-maps-set $\mathcal{R}_{3}=\{r_{BC}(X)|\forall X\in\ell^{OI}\}$, for a different fixed points $B,C$ in $\ell^{OI}-$line, forms a skew-field with ’addition and multiplication’ of points in $\ell^{OI}-$line. This, skew field $(\mathcal{R}_{3},+,\cdot)$ is sub-skew field of the skew field $(\ell^{OI},+,\cdot)$.