Engineering Calculus Notes 132

Engineering Calculus Notes 132 - P is any non-horizontal...

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120 CHAPTER 2. CURVES The product of the segments on a chord equals the product of the segments on the diameter perpendicular to it. B C Q R V Figure 2.2: Elements , Book VI, Prop. 13 In Figure 2.2 , this means | QV | 2 = | QV | · | V R | = | BV | · | V C | . (2.2) Conic Sections: Now consider the intersection of a plane P with the conical surface K . If P contains the origin A , there are three possible forms for the intersection P ∩ K : just the origin if P is horizontal or is tilted not too far o± the horizontal; a single generator if P is tangent to the cone, and a pair of generators otherwise. These are rather uninteresting. To classify the more interesting intersections when P does not contain the origin A , recall that P ∩ K is a circle when P is horizontal; so suppose that
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Unformatted text preview: P is any non-horizontal plane not containing A , and let γ be the intersection of P with K . Rotating our picture about the axis if necessary, we can assume that P intersects any horizontal plane in a line parallel to the x-axis. The yz-plane intersects P in a line that meets γ in one or two points; we label the ²rst P and the second (if it exists) P ′ ; these are the vertices of γ (Figure 2.3 ). Given a point Q on γ distinct from the vertices, let H be the horizontal plane through Q , and de²ne the points R , V , B and C as in Figure 2.2 . The line segments QV and PV are, respectively, the ordinate and abcissa ....
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