question archive 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
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").
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Answer:
(a) 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) 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) Hyperbolic
As it is possible to have more than one parallel (non-intersecting "line" [circle]) through a point that is not on a given "line" (circle).
(d) All the above axioms of incidence geometry are satisfied.
The elliptic parallel property holds. Any two great circles intersect. Hence there do not exist parallel lines.
