1. Consider the following models of finite geometries. You may assume that cach dark circle is a point and cach line that looks like a line is a line in the model. Determine which of the following models are Incidence Geometries, Projective Planes, and/or Alline Planes. Be sure to clearly explain your answor, NN Consider the infinite geometry ?, where points arc (z,y) K and lines are ar + by + =10 with a, b, | but not both a and b zero at the same time. Show that the geometric model B2, as delined above, is an Afline plance. We already verified the incidence axioms in class, so you just need to verily (P7) is satisfied. Note, verilying axioms in infinite models is a special type of prool. Do not just give examples! (This is similar to exercises 6.2 & 6.5 in your texthook. ) Prove that every Projective Plane is also an Incidence Geometry, (Hint: both the incidence & projective axioms arc on the section 6 handout as well as in vour textbook.) Consider the following interpretation of a geometry, Begin with a punctured sphere in Enclidean d-space. This is a sphere with one point, P, removed, where everything else about the sphere \"looks normal\". Let *points\" be points in the normal sense on the surface of the punctured sphere. Let \"straight lines\" be defined as circles on the surface of the sphere that pass through the point P (note: these are the only \"infinite straight lines\" for this infinite geometric model). Is this infinite model an incidence geometry? I so, does Playfair's Axiom hold in this model? Why or why not. Is this model isomorphic to any other geometric models we know? (Hint: a punctured sphere is often transformed by stereographic projection into a familar shape that is casier to work with. )