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Engineering Calculus Notes 58

Engineering Calculus Notes 58 - Book IV of the Elements...

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46 CHAPTER 1. COORDINATES AND VECTORS A B C D P Figure 1.25: Barycentric Coordinates The numbers λ i are called the barycentric coordinates of P with respect to A , B and C . Show that P lies on an edge of the triangle precisely if one of its barycentric coordinates is zero. Barycentric coordinates were introduced (in a slightly different form) by August M¨obius (1790-1860)) in his book Barycentrische Calcul (1827). His name is more commonly associated with “M¨obius transformations” in complex analysis and with the “M¨obius band” (the one-sided surface that results from joining the ends of a band after making a half-twist) in topology. 8 10. Find a line that lies entirely in the set defined by the equation x 2 + y 2 z 2 = 1. History note: 11. Heath [ 27 , pp. 85-88] points out that the proof of Proposition 4,
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Unformatted text preview: Book IV of the Elements contains the following implicit proof of Theorem 1.3.3 (see Figure 1.26 ). This was proved by vector methods in Exercise 8 . (a) The lines bisecting ∠ B and ∠ C intersect at a point D above BC , because of Book I, Postulate 5 (known as the Parallel Postulate ): That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced 8 The “M¨obius band” was independently formulated by Johann Listing (1808-1882) at about the same time—in 1858, when M¨ obius was 68 years old. These two are often credited with beginning the study of topology. [ 31 , p. 1165]...
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