Consider the following models, which provide interpretations of \"line\", \"poin
ID: 2899964 • Letter: C
Question
Consider the following models, which provide interpretations of "line", "point", and "incidence". In each of them, determine which of the axioms of incidence geometry hold, and if the model satisfies any of the parallel properties (elliptic, hyperbolic, or Euclidean).
a. "Points" are lines in Euclidean 3D space. "Lines" are planes in 3D space. "Incidence" is the usual relation of a line lying on a plane.
b. Same as in part (a), except now we restrict ourselves to lines and planes that pass through a fixed point O.
c. Fix a circle in the Euclidean plane. "Point" means a point inside the circle and "line" means a chord of the circle. "Incidence" means that the point lies on the chord.
d. Fix a sphere in Euclidean 3D space. Two points on the sphere are antipodal if they lie on a diameter of the sphere. For example, the north and south poles of the Earth are antipodal. Let "point" mean a set {P,P} of antipodal points, and let "line" mean a great circle on the sphere. Let the "line" C be incident with the "point" {P,P} if C contains both P and P.
Explanation / Answer
Euclidean 3D space.
For each pair of axioms of incidence geometry, invent an interpretation in which those two axioms are satisfied, but the third axiom is not. I-1, I-2 satisfied, I-3 not: Take a geometry for which there are no lines, and no points. Since there are no points, I-1 is satisfied, because one cannot find 2 distinct points. Similarly I-2 is satisfied, because there does not exist any line. I-3 is not satisfied because I-3 tells us that there exist at least 3 points, and there are none in our model. Another example is: The geometry with 1 point and no lines. Yet another example: Take the geometry with n points, where n 2, and one line ` which is incident with all points. Through every two distinct points there is a unique line, namely `. So I-1 is satisfied. Every line contains at least 2 points, so I-2 is satisfied. But I-3 is not satisfied because every 3 points lie on the line `, so one cannot find 3 points which are not collinear. I-1, I-3 satisfied, but not I-2: Let A, B, C be the points, and the lines are {A, B}, {A, C}, {B, C}, {A}. Clearly A, B, C do not lie on a line so I-3 is satisfied. The unique line through A, B is {A, B} and similarly there are unique lines through B and C and through A and C. So I-1 is satisfied. But I-2 is not satisfied because the line {A} does not contain 2 distinct points. I-2,I-3 satisfied, but not I-1: Take the geometry with 3 points A, B, C and no lines. Then I-2 and I-3 are clearly satisfied, but I-1 is not because there is no line through A and B.
If P and Q are distinct points then {P, Q} is a line. Since every line has 2 points, the only line containing P and Q is {P, Q}. So {P, Q} is the unique line through P and Q and I-1 is satisfied. Both in Example 3 and Example 4, the points A, B, C are distinct and not colinear, so I-3 is satisfied. In example 3, if we take the line {A, B} and the points C which does not lie on {A, B}, then there is a unique line through C parallel to {A, B} namely {C, D} (the other lines through C are {A, C}, {B, C} but they intersect {A, B}). By relabeling we see that for every line ` and 1 2 HARM DERKSEN every point P, not on ` there exists a unique line m which is parallel to ` and goes through P. The Euclidean parallel property is satisfied.