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# Spherical Geometry vs. Euclidean Geometry

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## Nathaniel Leon

on 1 December 2013

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#### Transcript of Spherical Geometry vs. Euclidean Geometry

What Uses Has Spherical Geometry Had on Today's Society?
Today, the concepts of spherical geometry are implemented in air and space travel, naval cruises, and much more. Spherical Geometry is used by pilots and ship captains as they navigate around the world. For example, an airplane looking to travel from Florida to the Philippines would pass over Alaska. Since the Philippines lie south of Florida, it does not seem reasonable to take this flight route. Yet, this happens to be the shortest distance between the two points, since Florida, Alaska, and the Philippines lie relatively “collinearly” along the path of a great circle. Thus, the best path to travel from Florida to the Philippines would include a flight route over Alaska. Spherical Geometry also aided navigators in mapping out the land and water. Last, spherical geometry was used in the the invention of the GPS (Global Positioning System).
Spherical Geometry vs. Euclidean Geometry
by Nathaniel Leon and Aaron Havard

What is Spherical Geometry?
Spherical geometry is an example of a geometry which is not Euclidean. It is the study of figures on the surface of a sphere (such as the spherical triangle and spherical polygon), as opposed to the type of geometry studied in Euclidean Geometry.
Who Created this Wonderful Form of Geometry?
There has been much debate over the past several hundred years about who really invented/discovered spherical geometry. However, some influential thinkers in the area of spherical geometry include:
Menelaus of Alexandria, a Greek mathematician and astronomer
Ibn Muʿādh al-Jayyānī, an Islamic mathematician and astronomer
Bernhard Riemann, a German mathmetician
Major Postulates and Theorems of Spherical Geometry
Major Postulates and Theorems of Euclidean Geometry
Euclidean Geometry is the study of geometry based on definitions, undefined terms, and the assumptions of the mathematician Euclid (330 B.C.). Euclidean Geometry is the study of flat space.
What is Euclidean Geometry?
Euclidean Geometry uses a plane to plot points and lines, whereas Spherical Geometry uses spheres to plot points and great circles.
Comparison of a Plane
On the sphere, points are defined as the usual sense--points on a sphere and points on a plane are the same. These points have no dimension and have no volume, area, or mass. Furthermore, as we will show and explain next, a line can be drawn between any two points.
Comparisons of a Point
The equivalents of lines are not defined in the usual sense of "straight line" in Euclidean geometry but in the sense of "the shortest paths between points," which are called geodesics. On the sphere the geodesics are the great circles; other geometric concepts are defined as in plane geometry but with straight lines replaced by great circles. In Euclidean Geometry, a straight line is infinite, whereas in Spherical Geometry, a great circle is finite and returns to its original starting point.
Comparisons of a Line
In spherical geometry angles are defined between great circles. We define the angle between two curves to be the angle between the tangent lines.
All angles will be measured in radians.
The sum of the interior angles of a triangle ALWAYS exceeds 180 degrees.
Comparison of Angle Measures
In Euclidean Geometry, two lines that intersect form exactly one point. However, in Spherical Geometry, when there are two great circles, they form exactly two intersecting points.
Comparison of Intersections of Lines
In Euclidean Geometry, perpendicular lines are formed when two lines are placed perpendicularly to each other. Perpendicular lines form four right angles and intersect at one point. In Spherical Geometry, perpendicular lines form to make eight right angles and intersect at two points.
Comparison of Perpendicular Lines
In Euclidean Geometry, parallel lines are existent. However, parallel lines are defined as never intersecting. Therefore, in Spherical Geometry, Then there are no parallel lines because any two "lines" (great circles) drawn on the sphere will intersect in two places.
Comparison of Parallel Lines