Projective plane over Ring/Division Ring/Field

Proportionality class

Proportionality class is a set of triples (x1,x2,x3) of reals in which ordinates preserves proportionality. Proportionality class of (x1,x2,x3) is denoted by [𝑥1,𝑥2,𝑥3 ]

Example:

The proportionality class [1,2,3] is a set given as below.

                [1,2,3]={(x,y,z) ∶ (1,2,3)=k(x,y,z)  ; k∈R-{0}}

In particular,

                [1,2,3]={(1,2,3),(2,4,6),(-1,-2,-3),(1/2,1,3/2)…}

                [2,4,6]={(2,4,6),(1,2,3),(-1,-2,-3),(1/2,1,3/2)…}

Thus,     [1,2,3]=[2,4,6], However, (1,2,3)≠ (2,4,6)

Video 1

Video 2

Video 3

Projective plane over Ring/Division Ring     

Let D be a division ring and T = D × D× D/(0,0,0) be a set of all non-zero triples of D.

Also we say (x₁,x₂,x₃) and (y₁,y₂,y₃) in T are left proportional if there exist k∈D such that x_i=ky_i. Similarly, we say (l₁,l₂,l₃) and (m₁,m₂,m₃)  in T are right proportional if there exist k∈D such that  l_i=km_i for i=1,2,3

Now define

            P_D = {[x₁,x₂,x₃]: (x₁,x₂,x₃)∈T}

            L_D = {<l₁,l₂,l₃>: (l₁,l₂,l₃)∈T}

            I_D = {({[x₁,x₂,x₃], <l₁,l₂,l₃>): x₁l₁+x₂l₂+x₃l₃=0}

Then π _D =(P_D,L_D,I_D) is incidence structure, called plane over D.

Theorem

Prove that π _D =(P_D,L_D,I_D)  is projective plane.

Since, π _R is projective plane satisfying the statements

(1)   Each point is proportionality class [x₁,x₂,x₃] of triples of real numbers not all zero

(2)   Each line is proportionality class <l₁,l₂,l₃> of triples of real numbers not all zero

(3)   Point <l₁,l₂,l₃> is on line <l₁,l₂,l₃> if and only if x₁l₁+x₂l₂+x₃l₃=0

Now, we need to show that π _D ~ π _R, for this

      (1)   P_D = {[x₁,x₂,x₃]: (x₁,x₂,x₃)∈T}

Each point [x₁,x₂,x₃] is left proportionality class of triples (x₁,x₂,x₃) not all zero

            Implies statement 1 of π _R

      (2)   L_D = {<l₁,l₂,l₃>: (l₁,l₂,l₃)∈T}

Each line <l₁,l₂,l₃>  is right proportionality class of triples (l₁,l₂,l₃) not all zero

Implies statement 2 ofπ _R

       (3)   I_D = {({[x₁,x₂,x₃], <l₁,l₂,l₃>): x₁l₁+x₂l₂+x₃l₃=0}

The point [x₁, x₂, x₃] is on the line <l₁,l₂,l₃> if and only if

       x₁l₁+x₂l₂+x₃l₃=0

Implies statement 3 of π _R

Hence the theorem.

Projective plane over Field

Let F be a division ring and T = F × F× F/(0,0,0) be a set of all non-zero triples of F.

Then

            P_F = {[x₁,x₂,x₃]: (x₁,x₂,x₃)∈T}

            L_F = {<l₁,l₂,l₃>: (l₁,l₂,l₃)∈T}

            I_F = {({[x₁,x₂,x₃], <l₁,l₂,l₃>): x₁l₁+x₂l₂+x₃l₃=0}

Then π _F =(P_F,L_F,I_F) is incidence structure, called plane over F.

Here, π _F =(P_F,L_F,I_F) is a projective plane over F.

Desarguesian Palne (Couple)

The French mathematician Gerard Desargues (1593–1662) was one of the earliest contributors to the study of Projective Geometry. A theorem known as “Desargues theorem” bears his name and often set him as inventor of Projective Geometry. This theorem is very useful in the development of Projective Geometry. It is about relating two aspects of a pair of triangles: pair of three vertices and pair of three sides. We deal this theorem using terms as below.

Couple

Two triangles Δabc and Δa'b'c' are said to be a couple if

                Points a,b,c,a',b',c' are distinct

                Lines aa', bb',cc' are distinct

                Lines aa', bb',cc' does not lie on either sides of triangles.

Unit 3 PG# Introduction

3.1.  Introduction

The French mathematician Gerard Desargues (1593–1662) was one of the earliest contributors to the study of Projective Geometry. A theorem known as “Desargues theorem” bears his name and often set him as inventor of projective geometry. This theorem is very useful in the development of projective geometry. It is about relating two aspects of projective geometry: pair of three vertices and pair of three sides. We deal this theorem with a couple of terms as below.

Couple

Two triangles Δabc and Δa'b'c' are said to be a couple if

                Points a,b,c,a',b',c' are distinct

                Lines aa', bb',cc' are distinct

                Lines aa', bb',cc' does not lie on either sides of triangles.

Central Couple

Two triangles Δabc and Δa'b'c' are said to be a central couple if

                Δabc  and Δa'b'c' are couple

                lines  aa',bb',cc' are concurrent
NOTE

• The point of concurrency is called center of the couple
• Two triangles which are perspective from a point is called central couple.

Axial Couple

Two triangles Δabc and Δa'b'c' are said to be an axial couple if

·         Δabc and Δa'b'c' are couple

·         points ab∩a'b', bc∩c'c', ca∩c'a' are collinear.
       NOTE

• The line of collinearity is called axis of the couple.
• Two triangles which are perspective from a line is called axial couple

3.2.  Desargues triangle theorem

If two triangles are so situated that line joining the pair of corresponding vertices are concurrent then the points of intersection of pair of corresponding sides are collinear and conversely. This theorem can be stated as “if two triangles are central couple then they are axial couple and conversely”. The Desargues theorem can also be stated as “if two triangles are perspective from a point then they are perspective from a line and conversely”. In the figure below, Δabc and Δa'b'c' satisfy Desargues triangle theorem.

Desarguesian Plane

A projective plane in which Desarguesian triangle theorem holds is called Desarguesian plane.