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Engineering Calculus Notes 131

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

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2.1. CONIC SECTIONS 119 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 axis 6 ) 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 = m 2 ( x 2 + 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
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