In each of the following interpretations of the undefined terms, which of the ax

Question

In each of the following interpretations of the undefined terms, which of the axioms of incidence geometry are satisfied and which are not? Tell whether each interpretation has the elliptic, Euclidean, or hyperbolic parallel property. Consider this as an informal problem, in the sense that your explanations are allowed to be informal and aided by illustrations.

(a) ”Points” are lines in R3 and ”lines” are planes in R3, incidence is the usual relation of a line lying in a plane.

(b) Same as in (a), except that we restrict ourselves to lines and planes that go through a fixed point O.

(c) This was discussed in class. Just make sure you remember. Fix a circle in R2 and take the points as the points in the interior of the circle and lines to be chords of the circle, and let incidence be the usual of a point lying on a chord.

(d) Fix a sphere in R3. two points are called antipodal, if they lie on a diameter of the sphere, e.g. the north and south pole are antipodal points. Interpret a ”point” to be a set {P,P} consisting of two points on the sphere that are antipodal, and interpret a ”line” to be a great circle on the sphere. Interpret a point {P,P} to ”lie on” the line C, if both P and P lie on C - actually, if one of them lies on C, then the other one does also, since C is a great circle.

Explanation / Answer

(a) "Points" are lines in the Euclidean three-dimensional space, "lines" are planes in the Euclidean three-space, "incidence" is the usual relation of a line lying on a plane.

Ans: Euclidean

Given a "line" (plane in 3D) and a "point" (line) not on it, there is only one "line" (plane) that does not intersect the given one - namely the plane that is parallel in 3D.

(b) Same as in part (a), except that we restrict ourselves to lines and planes that pass through fixed point, O

Ans: elliptic

If each "line" (plane) goes through O, then each "line" (plane) has an intersection with every other "line" (plane), as they must have O in common. So the two either are the same, or they intersect "properly".

(d) Fix a sphere in Euclidean three-space. Two points on the sphere are called antipodal if they lie on a diameter of the sphere; e.g., the north and south poles are antipodal. Inperpret a "point" to be a set {P,P'} consisting of two points on the sphere that are antipodal. Interpret a "line" to be a great circle on the sphere. Inperpret a "point" {P,P'} to "lie on" a "line" C if both P and P' lie on C (actually, if one lies on C, then so does the other, by definition of "great circle").

Ans: All the above axioms of incidence geometry are satisfied

c) The elliptic parallel property holds. Any two great circles intersect. Hence there do not exist parallel lines.

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