Question

3. Determine which of the following are models of Incidence Geometry. For those th are models, indicate which parallel property holds for the model. For those that a not a model, list at least one axiom that fails and illustrate why. a. Points are points in the Euclidean plane and lines are circles with positive radius. b. Points are in {(x, y) = R2 22 + y2 <9} and lines are open chords of the circle. c. Points are points in the Euclidean plane and lines are Euclidean lines that have slo m = 8. d. Points are points in {(x, y, z) e R3 | x2 + y2 + 22 < 9 and 2 >0} and lines are gre semicircles in this set. e. Points are the vertices of an octahedron and lines are all sets of exactly two points

Answer 1

Notice that for any two distinct points and in the Euclidean plane there exists more than one circle on which they lie, so the first axiom of incidence geometry i.e., For every distinct point and there is a unique line incident with P and Q, is not satisfied here. Hence, this is not a model of incidence geometry.

All the three axioms of incidence geometry is satisfied hence this is a model of incidence geometry.

Euclidean parallel property hold for this model.

First axiom of incidence geometry is not satisfied because, there are pairs of distinct points that do not lie on any line.

First axiom of incidence geometry is not satisfied because, there are pairs of distinct points that do not lie on any line.

It satisfies all the three axioms of incidence geometry. And since for any line   and is a point not lying on , there exist two distinct lines through that are parallel to .

Hence Hyperbolic parallel property holds for this model.