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

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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 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.

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

(c) Fix a circle in the Euclidean plan. 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.

(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. Interpret 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").