Engineering Calculus Notes 131

Engineering Calculus Notes 131 - 119 2.1. CONIC SECTIONS...

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Unformatted text preview: 119 2.1. CONIC SECTIONS Here, we give a simplified and anachronistic version of the basic ideas in Book I, bowlderizing [25, pp. 355-9]. Conical Surface: Start with a horizontal circle C ; on the vertical line through the center of C (the axis6 ) pick a point A distinct from the center of C . The union of the lines through A intersecting C (the generators) is a surface K consisting of two cones joined at their common vertex (Figure 2.1). If we put the origin at A, the axis Figure 2.1: Conical Surface K coincides with the z -axis, and K is the locus of the equation in rectangular coordinates z 2 = m2 (x2 + y 2 ) (2.1) where m = cot α is the cotangent of the angle α between the axis and the generators. Horizontal Sections: A horizontal plane H not containing A intersects K in a circle centered on the axis. The yz -plane intersects H in a line which meets this circle at two points, B and C ; clearly the segment BC is a diameter of the circle. Given a point Q on this circle distinct from B and C (Figure 2.2), the line through Q parallel to the x-axis intersects the circle in a second point R, and the segment QR is bisected at V , the intersection of QR with the yz -plane. A basic property of circles, implicit in Prop. 13, Book VI of Euclid’s Elements [27, vol. 2, p. 216] and equivalent to the equation of a circle in rectangular coordinates (Exercise 3), is of Eutocius’ edition by Thabit ibn Qurra (826-901); the eighth book is lost. A modern translation of Books I-IV is [41]. An extensive detailed and scholarly examination of the Conics has recently been published by Fried and Unguru [15]. 6 Apollonius allows the axis to be oblique —not necessarily normal to the plane of C . ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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