I have a question that needs your help In each of the follow

I have a question that needs your help.

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; e.g., the north and south poles are antipodal. Interpret a \"point\" to be a set IP, P\'h consisting of two points on the sphere that are antipodal. Interpret a \"line\" to be a great circle on the sphere

Solution

The following axioms set out the basic incidence relations between lines, points and planes. They also characterise the concept of ``dimension\'\' that we associate with these notions.

All three incidence axioms are satisfied here. In a model, meaning to the undefined terms are given. From there we can check whether or not the axioms are true.

a. For 1 this one get translated into \"given any two distinct lines through the origin, they lie on a unique plane through the origin.\" This is considered to be true.

For 2, this one get translated into \"any plane through the origin contains at least two distinct lines through the origin\" This is also true. There exist many such lines (infinitely)

For 3 , this is translated to \"there exist three distinct lines through the origin which do not lie on the same plane\". This one is also true. Here we can take the example of three coordinates axes.

Therefore, this model has the euclidean property , because any two distinct lines which are distinct planes through the origin intersect in a line which is a point in our model. so, no parallel lines are there.

b.  It is Elliptic Property because, if each \"line\" ( or plane) goes through O, then each \"line\" (or plane) has an intersection with every other \"line\" (or plane), as they must have O in common. So either the two will be or they will intersect properly.

c.All the axioms of incidence geometry are satisfied.

d. Here the elliptic parallel property is there as any two circles will intersect. Therefore there are no parallel lines.