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Constructions in Projective Geometry
Bernhard Werner
31. Juli 2016
Bernhard Werner Constructions in Projective Geometry 31. Juli 2016 1 / 37
Disclaimer
No information in this course is guaranteed to be correct. References tothis course in the exam are NOT valid justifications for your answers.
Axiomatic constructions
Projective planes from affine planes
General procedure:
1 Add 1 new point for each pencil of parallel lines.
2 Create 1 new line on which all those points lie.
General procedure:
1 Add 1 new point for each pencil of parallel lines.
2 Create 1 new line on which all those points lie.
The Fano-plane F2P2.
Projective planes from affine planesThe plane F3P2.
Basic constructions
Join & Meet
Constructions:
Computing in RP2:Given points A,B,C ,D. What is (A B) (C D)?
(A B) (C D)
or
[C ,D,B ] A [C ,D,A] B.
Join & Meet
Constructions:
(A B) (C D)
or[C ,D,B ] A [C ,D,A] B.
Join & Meet
Constructions:
(A B) (C D)
Join & Meet
Constructions:
(A B) (C D)
Join & Meet
Constructions:
(A B) (C D)
Join & Meet
Constructions:
(A B) (C D)
The simplest construction
Given a point P and a line l . How to construct the parallel to l through P?
Incidence theorems
Theorem of Desargues
Theorem of Pappos
Theorem of Pappos
Theorem of Pappos
Theorem of Pappos
Theorem of Pappos
Theorem of Pappos
Theorem of Pappos
Theorem of Pappos
Theorem of Pappos
Scissor theorem
Scissor theorem
Scissor theorem
Scissor theorem
Scissor theorem
Constructions in RP1
Cross-ratio
We identify(x
1
) RP1 with x R and
(10
) RP1 with
(1
)and
therefore with .
Some important identities:
We have an isomorphism RP1 R {} via P 7 (0,;P, 1).Specifically, for x R {} we have (0,; x , 1) = x .For (A,B;C ,D) = we have
(B,A;C ,D) = 1 and(A,C ;B,D) = 1 .
All combinations of these permutations lead to the values
,1, 1 , 1
1 , 11
and
1 .
Cross-ratio
We identify(x
1
) RP1 with x R and
(10
) RP1 with
(1
)and
therefore with .Some important identities:
We have an isomorphism RP1 R {} via P 7 (0,;P, 1).Specifically, for x R {} we have (0,; x , 1) = x .For (A,B;C ,D) = we have
(B,A;C ,D) = 1 and(A,C ;B,D) = 1 .
All combinations of these permutations lead to the values
,1, 1 , 1
1 , 11
and
1 .

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