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MAT 360 Lecture 4

MAT 360 Lecture 4. Projective and affine planes. Undefined terms. Point Line Incidence. Incidence Axioms. For each point P and for each point Q not equal to P there exists a unique line incident with P and Q. For every line T there exist at least two distinct points incident with T.

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MAT 360 Lecture 4

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  1. MAT 360 Lecture 4 • Projective and affine planes

  2. Undefined terms • Point • Line • Incidence

  3. Incidence Axioms • For each point P and for each point Q not equal to P there exists a unique line incident with P and Q. • For every line T there exist at least two distinct points incident with T. • There exist three distinct points with the property that no line is incident with all the three of them.

  4. Recall • Two lines l and m are parallel if no point lies on both or them.

  5. Definition • Euclidean Parallel Property: For every line l and for every point P not in l there exists exactly one line parallel to l passing through P.

  6. Affine Plane Axioms (Incidence+ Euclidean Parallel Property, (EPP)) • For each point P and for each point Q not equal to P there exists a unique line incident with P and Q. • For every line l there exist at least two distinct points incident with l. • There exist three distinct points with the property that no line is incident with all the three of them. • For every line l and for every point P not in l there exists exactly one line parallel to l passing through P. (EPP)

  7. EXERCISE • Can you give a model for the Affine Plane Axioms?

  8. EXERCISE: Is the following a model for the Affine Plane Axioms? • Points : A, B, C • Lines {A,B}, {B,C}, {A,C}, • Incidence: Set membership.

  9. EXERCISE: Is the following a model for the Affine Plane Axioms? • Points : A, B, C, D • Lines {A,B}, {B,C}, {C,D}, {A,C}, {A,D},{B,D} • Incidence: Set membership.

  10. EXERCISE: Is the following a model for the Affine Plane Axioms? • Points : A, B, C, D, E, F, G • Lines {A,B,D}, {A,C,G}, {G,E,D}, {G,F,B}, {C,F,D},{E,F,A},{C,B,E} • Incidence: Set membership.

  11. Definition • Elliptic Parallel Property: For every pair of lines l and m there exist a point P incident with both of them.

  12. Is at least one of these two statements valid in every model of Incidence Geometry? • Elliptic Parallel Property: For every pair of lines l and m there exist a point P incident with both of them. • Euclidean Parallel Property: For every line l and for every point P not in l there exists exactly one line parallel to l passing through P.

  13. Parallel properties • Elliptic Parallel Property • Euclidean Parallel Property • Hyperbolic Parallel Property: For every line l and for every point P not in l there are at least two lines incident with P and parallel to l.

  14. Projective Plane Axioms • For each point P and for each point Q not equal to P there exists a unique line incident with P and Q. • For every line l there exists at least three distinct points incident with l. • There exist three distinct points with the property that no line is incident with all the three of them. • For every pair of lines l and m there exist a point P incident with both of them.

  15. EXERCISE: Is the following a model for the Projective Plane Axioms? • Points : A, B, C, D • Lines {A,B}, {B,C}, {C,D}, {A,C}, {A,D},{B,D} • Incidence: Set membership. • If the above is not a model, study which, if any, of the projective plane axioms hold

  16. EXERCISE: Is the following a model for the Projective Plane Axioms? • Points : A, B, C • Lines {A,B}, {B,C}, {A,C}, • Incidence: Set membership. • If the above is not a model, study which, if any, of the projective plane axioms hold

  17. EXERCISE: Is the following a model for the Projective Plane Axioms? • Points : A, B, C, D, E, F, G • Lines {A,B,D}, {A,C,G}, {G,E,D}, {G,F,B}, {C,F,D},{E,F,A},{C,B,E} • Incidence: Set membership. • If the above is not a model, study which, if any, of the projective plane axioms hold

  18. Definition • A projective plane is model of the Projective Plane Axioms

  19. A recipe to construct a projective plane from an affine plane A. • On the set of lines of A, define the equivalence relation: The line l is equivalent to the line m if l and m are parallel or equal. • Define new model B. • The points of B are the points in A and the equivalence classes defined in 1 in this recipe. • For each line l in A, define a line in B with all the points of l and the equivalence class determined by l • Add one more line to B, which is the union of all the equivalence classes of 1.

  20. EXERCISE: Apply the recipe to the affine plane given by, • Points : A, B, C, D • Lines {A,B}, {B,C}, {C,D}, {A,C}, {A,D},{B,D} • Incidence: Set membership.

  21. EXERCISE: Check that the recipe produces a projective plane.

  22. Homework for next Tueday • Chapter 2: • Exercise 9 (20 points), • Exercise 10 (20 points), • Exercise 11. • Extra credit: Exercise 12. • (Remember to justify your answers.)

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