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. (a) "Points" are lines in Euclidean three-dimensional space, "lines" are planes in Euclidean three-space, "incidence" is the usual relation of a line lying in a plane. (b) Same as in part (a), except that we restrict ourselves to lines and planes that pass through a fixed point O. (c) Fix a circle in the Euclidean plane. Interpret "point" to mean a Euclidean point inside the circle, interpret "line" to mean a chord of the circle, and let "incidence" mean that the point lies on the chord. (A chord of a circle is a segment whose end- points lie on the circle.) (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: eg. the north and south poles are antipodal. Interpret 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.

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".

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

ans so its elliptic

(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

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