Models and Interpretations of Incidence Geometry

EXAMPLES:

Interpretation # 1

Points:  A, B, and C. Lines: {A, B}, {B, C}, and {A, C}. Incidence: P_I l if P is an element of l.

Interpretation # 2

Points:  A, B, and C. Lines: {A}, {A, B}, {B, C}, and {A, C}. Incidence: P_I l if P is an element of l.

Interpretation # 3

Points:  A, B, and C. Lines: {A, B}, and {A, C}. Incidence: P_I l if P is an element of l.

Interpretation # 4

Points:  A, B, C, and D. Lines: {A, B}, {B, C}, {C, D}, {A, D}, {B, D} and {A, C}. Incidence: P_I l if P is an element of l.

Interpretation # 5

Points:  A, B, and C. Lines: {A, B, C}. Incidence: P_I l if P is an element of l.

Interpretation # 6

Points:  {A, B}, {B, C}, and {A, C}. Lines: A, B, and C. Incidence: P_I l if l is an element of P.

Interpretation # 7

Points:  A, B, C, D, and E. Lines: {A, B}, {A, C}, {A, D}, {A, E}, {B, E}, {C, E}, {D, E}, and {B, C, D}. Incidence: P_I l if P is an element of l.

Interpretation # 8

Points: {A}, {A, B}, and {A, B, C}. Lines: A, B, and C. Incidence: P_I l if  l is an element of P.

Interpretation # 9

Points:  A, B, C, D, and E. Lines: {A, B}, {A, C}, {A, D}, {A, E}, {B, C}, {B, D} {B, E}, {C, D}, {C, E}, and {D, E}. Incidence: P_I l if P is an element of l.

Interpretation # 10

Points: all dots on a sheet of paper. Lines: all horizontal and vertical lines. Incidence: P_I l if the dot P  lies on line l.

QUESTIONS:

Does this interpretation satisfy the incidence axiom I-1?

Does this interpretation satisfy the incidence axiom I-2?

Does this interpretation satisfy the incidence axiom I-3?

Is this interpretation a model?

Determine whether this interpretation has the elliptic, Euclidean, of hyperbolic parallel property.

ANSWERS:

To find the answer to the questions above and an  explanation for a specific interpretation, click on that interpretation. Also, the following table summarizes all the answers: