Comparing and Contrasting Euclidean, Spherical, and Hyperbolic Geometri

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.   Ã‚  Ã‚  Ã‚  Ã‚  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 â€Å"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 â€Å"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 â€Å"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, rigid-motion-along-itself symmetry, and central symmetry. If we find that a line on a sphere satisfies all of the above condition, then that line is straight on a sphere. I have included my homework assignment for straightness on a sphere so that one can see why a straight line on a sphere must satisfy these conditions. Finally, I need to give my definition of a straight line on a hyperbolic... ...h other along a third line, l. Then to consider the geometric figure that is formed by the three lines and look for the symmetries of that geometric figure. Then we were asked what we could say about the lines r and r’. I have provided my notes that include an outline to this proof for all three surfaces so that one can see the conclusions that we made as a class. We found that on a Euclidean plane parallel transported lines do not intersect and are equidistant. For a hyperbolic plane we found that parallel transported lines diverge in both directions. Finally for a sphere we found that parallel transported lines always intersect.   Ã‚  Ã‚  Ã‚  Ã‚  Using all the above material, we can see that there are many different similarities and differences when looking at a Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry. Using my artifacts will help one understand many of my conclusions about these three surfaces. This essay was an excellent opportunity to reflect on my growing understanding of these three surfaces. I hope you, the reader, can benefit from my conclusions and gain a better understanding of the similarities and differences of these three surfaces.