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, 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 plane. My definition of a straight line on a hyperbolic plane must satisfy 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, and self-symmetry. If a line on a hyperbolic plane satisfies these conditions then we can say that it is straight. I have included my homework of my definition of a straight line on a hyperbolic plane so that one can see why these conditions must be satisfied.

The next issue that I will address for these three geometries is the definition of an angle on all three surfaces. The definition that I will give applies to all three surfaces. There are at least three different perspectives from which we can define "angle". These include: a dynamic notion of angle-angle as movement, angles as measure, and angles as a geometric shape. A dynamic notion of angle involves an action which may include a rotation, a turning point, or a change in direction between two lines. Angles as measure may be thought of as the length of a circular arc or the ratio between areas of circular sectors. When thinking of an angle as a geometric shape an angle may be seen as the delineation of space by two intersecting lines. I have provided my homework assignment on my definition of an angle so that one can see the reasoning of my definition for all three surfaces. However, my homework assignment does not ask to define an angle on a hyperbolic plane. This is because a region on a hyperbolic plane can be looked at locally to have the same results as a Euclidean Plane. Since we are on the topics of angles I need to mention the Vertical Angle Theorem. In my homework I used two different proofs to prove the Vertical Angle Theorem on a Euclidean plane and a sphere. The first idea I used was looking at the Vertical Angle Theorem using angle as measure. The second idea I used was looking at the Vertical Angle Theorem using angle as rotation. I have provided my homework so that one can see my reasoning behind both of these proofs. I found that they worked on a Euclidean plane and a sphere. Although I did not have to say if my proofs worked on a hyperbolic plane, I can say that they would because we can look at a hyperbolic plane locally.

From Chapter 6 in our textbook Experiencing Geometry by Henderson and Taimina, we formulated a summary of the properties of geodesics on the plane, spheres, and hyperbolic planes. I feel this is a good homework assignment to mention in this paper. For the first part of the problem we were to explain why for every geodesic on the plane, sphere, and hyperbolic plane there is a reflection of the whole space through the geodesic. For the second part of the problem I showed that every geodesic on the plane, sphere, and hyperbolic plane can be extended indefinitely (in the sense that the bug can walk straight ahead indefinitely along any geodesic). The third part asked to show that for every pair of distinct points on the plane, sphere, and hyperbolic plane there is a (not necessarily unique) geodesic containing them. In the fourth part of the problem I showed the every pair of distinct points on the plane or hyperbolic plane determines a unique geodesic segment joining them On the sphere there are always at least two such segments. Finally, in the fifth part of the problem I showed that on the plane or on a hyperbolic plane, two geodesics either coincide or are disjoint or they intersect in one point. On a sphere, two geodesics either coincide or intersect exactly twice. I have provided the homework assignment as an artifact so that one can read through my explanations. Instead of rewriting them for this part of the essay I decided to just include my previous work.

I also discovered why the Isosceles Triangle Theorem (ITT) held on