Latitude lines, except for the equator, are not great circles. The longitude lines and the equator are great circles of the Earth. A great circle is the largest circle that can be drawn on a sphere. Therefore, lines in spherical geometry are great circles. In spherical geometry, points are defined in the usual way, but lines are defined such that the shortest distance between two points lies along them. In a plane geometry, the basic concepts are points and lines. Spherical geometry is a plane geometry on the surface of a sphere. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line. In spherical geometry there are no such lines. The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it. The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes. Non-Euclidean geometry is any geometry that is different from Euclidean geometry. It has been used by the ancient Greeks through modern society to design buildings, predict the location of moving objects and survey land. Any two straight lines equidistant from one another at two points are infinitely parallelĮuclidean geometry is of great practical value. Any center and any radius can describe a circleĥ. A straight line can be extended infinitely in either directionģ. Between any two points is a straight lineĢ. Most of the theorems which are taught in high schools today can be found in Euclid’s 2000 year old book.Įuclid had decreed five postulates which became the bases for geometry :ġ. His book, called “The Elements”, is a collection of axioms, theorems and proofs about squares, circles acute angles, isosceles triangles, and other such things. Euclidean geometry was named after Euclid, a Greek mathematician who lived in 300 BC. The geometry with which we are most familiar is called Euclidean geometry.
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