# Consider the following models, which provide interpretations of “line”, “point”, and “incidence”….

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’.

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