A History of Curvature and Applications of Hyperbolic Space

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While the study of curvature is an ancient one, the geometry of curved surfaces is a topic that has been slowly developed over centuries. The Ancient Greeks certainly considered the curvature of a circle and a line distinct, noting that lines do not bend, while circles do. Aristotle expanded on this concept explaining that there were three kinds of loci: straight, circular, and mixed (Coolidge)Then in the third century B.C. Apollonius of Perga found that at each point of a conic section there is exactly one normal line (Coolidge, 375-6). However, the Greeks had little to more to offer in the study of curvature.
In the fourteenth century, it was Nicolas Oresme who gave the a definition of curvature. Oresme defines “Curvitas” as follows: if there are two curves touching the same line at the same point then the smaller curve will have greater curvature (Coolidge, 376). After Oresme, there is a gap of nearly 300 years before Kepler began again the discussion of curvature. Kepler discovered that if one takes “a curve all of whose tangents are on one side. To this is attached a flexible string which is pulled taut and then unwound, the curve from which the string springs is called the evolute” (Coolidge, 377). From here, he provides the theorem that states if two curves have tangents on one side which have a common point cannot have the same set of normals (Coolidge (377). While his theorems because quite useful, his ignorance of the calculus did not allow him to extend his theorems to the general cases.
Then Sir Issac Newton in the 17th century worked to develop the concept of the center of curvature which is the center of a circle having the same curvature, and no other tangent circle can lie between this and the c...

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