n each of the following interpretations of the undefined terms, which of the axi
ID: 2899405 • Letter: N
Question
n each of the following interpretations of the undefined terms, which of the axioms or 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, and 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 points to mean chord of the circle, and let incidence mean that the pint 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. 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. Interpret a point {P, P} to lie on a line C if both P and P lie on C.
Explanation / Answer
(a) The “points” are lines through the origin, and “lines” are planes through the origin, with incidence of “point on line” being the usual incidence of line on plane. For 1 I.1: this gets translated to “given any two distinct lines through the origin, they lie on a unique plane through the origin.” This is true (can be defined using the cross product, if you wish.) For I.2: this gets translated to “any plane through the origin contains at least two distinct lines through the origin.” This is also true, in fact there are infinitely many such lines. For I.3: this gets translate to “there exist three distinct lines through the origin which do not lie on the same plane.” Also true, take for example the three coordinate axes. Finally, this model has the elliptic parallel property, because any two distinct “lines,” which are distinct planes through the origin, interesect in a line, which is a “point” in our model. So there are no parallel lines
(c) The “points” are points inside a circle, and the “lines” are straight segments whose endpoints are on the circle. In this case we are really taking just a subset of the x-y-plane, the interior of a circle. The three axioms I.1, I.2, and I.3 are trivially satisfied since they are satisfied in any piece of the x-y-plane. This model has the hyperbolic parallel property. Given a chord of a circle and a point P inside the circle which is not on the chord, there are infinitely many chords through P which do not intersect our given chord.
(d) This model is isomorphic to the model in part (a). Given a line through the origin, intersecting it with a sphere centred at the origin gives a pair of anti-podal points. This gives a one-to-one correspondence between lines through the origin and pairs of anti-podal points. Similarly, given a plane through the origin, intersecting it with the sphere gives a great circle (equator). Again, this is a one-to-one correspondence between planes through the origin and great circles. It is clear that these correspondences preserve the incidence relation. Hence this model is isomorphic to (a) and is a model for incidence geometry with the elliptic parallel property
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