1373 Words6 Pages

Differences in Geometry
Geometry is the branch of mathematics that deals with the properties of space. Geometry is classified between two separate branches, Euclidean and Non-Euclidean Geometry. Being based off different postulates, theorems, and proofs, Euclidean Geometry deals mostly with two-dimensional figures, while Demonstrative, Analytic, Descriptive, Conic, Spherical, Hyperbolic, are Non-Euclidean, dealing with figures containing more than two-dimensions. The main difference between Euclidean, and Non-Euclidean Geometry is the assumption of how many lines are parallel to another. In Euclidean Geometry it is stated that there is one unique parallel line to a point not on that line.
Euclidean Geometry has been around for*…show more content…*

Spherical Geometry is also the most commonly used Non-Euclidean geometry, being used by astronomers, pilots, and ship captains. In Euclidean geometry it is stated that the sum of the angles in a triangle are equal to 180. As for Spherical geometry it is stated that the sum of the angles in a triangle are always greater than 180. When most people try and visualize a triangle containing angle sums greater than 180 they say it's impossible. They're right, in Euclidean geometry it is impossible, but as for Spherical geometry, it is possible. Think of the triangle on a sphere, and then try and visualize it. See Appendix 1-1. When thinking of the Non-Euclidean Spherical Geometry, we start of with a basic sphere. A sphere is a set of points in three-dimensional space equidistant from a point called the center of the sphere. The distance from the center to the points on the sphere is called the radius. See Appendix 1-2 to visualize tangents, lines, and centers between the sphere, lines, and planes. Unlike standard Euclidean Geometry, in Spherical Geometry, radians are used to replace degree measures. It is usual for most people to measure angles and such with degrees, as for scientists, engineers, and mathematicians, radians are used to substitute degree measures. The size of a radian is determined by the requirement that there are 2pi radians in a circle. Thus 2pi radians equals 360 degrees. This means that 1 radian =

Spherical Geometry is also the most commonly used Non-Euclidean geometry, being used by astronomers, pilots, and ship captains. In Euclidean geometry it is stated that the sum of the angles in a triangle are equal to 180. As for Spherical geometry it is stated that the sum of the angles in a triangle are always greater than 180. When most people try and visualize a triangle containing angle sums greater than 180 they say it's impossible. They're right, in Euclidean geometry it is impossible, but as for Spherical geometry, it is possible. Think of the triangle on a sphere, and then try and visualize it. See Appendix 1-1. When thinking of the Non-Euclidean Spherical Geometry, we start of with a basic sphere. A sphere is a set of points in three-dimensional space equidistant from a point called the center of the sphere. The distance from the center to the points on the sphere is called the radius. See Appendix 1-2 to visualize tangents, lines, and centers between the sphere, lines, and planes. Unlike standard Euclidean Geometry, in Spherical Geometry, radians are used to replace degree measures. It is usual for most people to measure angles and such with degrees, as for scientists, engineers, and mathematicians, radians are used to substitute degree measures. The size of a radian is determined by the requirement that there are 2pi radians in a circle. Thus 2pi radians equals 360 degrees. This means that 1 radian =

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