Comparing and Contrasting Euclidean, Spherical, and Hyperbolic Geometries

When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for Euclidean Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where something is true for one or two geometries but not the other geometry. However, sometimes a property is true for all three geometries. These points bring us to the purpose of this paper. This paper is an opportunity for me to demonstrate my growing understanding about Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry. The first issue that I will focus on is the definition of a straight line on all of these surfaces. For a Euclidean plane the definition of a "straight line" is a line that can be traced by a point that travels at a constant direction. When I say constant direction I mean that any portion of this line can move along the rest of this line without leaving it. In other words, a "straight line" is a line with zero curvature or zero deviation. Zero curvature can be determined by using the following symmetries. These symmetries include: reflection-in-the-line symmetry, reflection-perpendicular-to-the-line symmetry, half-turn symmetry, rigid-motion-along-itself symmetry, central symmetry or point symmetry, and similarity or self-similarity "quasi symmetry." So, if a line on a Euclidean plane satisfies all of the above conditions we can say it is a straight line. I have included my homework assignment of my definition of a straight line for a Euclidean plane so that one can see why I have stated this to be my definition. My definition for a straight line on a sphere is very similar to that on a Euclidean Plane with a few minor adjustments. My definition of a straight line on a sphere is one that satisfies the following Symmetries. These symmetries include: reflection-through-itself symmetry, reflection-perpendicular-to-itself symmetry, half-turn symmetry,