Engineering Calculus Notes 134

Engineering Calculus Notes 134 - 122 CHAPTER 2 CURVES The...

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Unformatted text preview: 122 CHAPTER 2. CURVES The coefficient p above is called the parameter of ordinates for γ . • Ellipses: If P V is not parallel to AC , then the line P V (extended) meets the line AB (extended) at a second vertex P ′ . If φ denotes the (acute) angle between P and a horizontal plane H, then V lies between P and P ′ if 0 ≤ φ < π − α and P lies 2 between V and P ′ if π − α < φ ≤ π . 2 2 In the first case, in contrast to the case of the parabola, the ratio of |QV |2 to |P V | depends on the point Q on the curve γ . To understand it, we again form the “orthia” of γ as a line segment P L perpendicular to P V with length p. Now let S be the intersection of LP ′ with the line through V parallel to P L (Figure 2.4). A P V C J B L S H P′ Figure 2.4: Definition of S One derives the equation (see Appendix A) |QV |2 = |V S | · |P V | . (2.5) This is like Equation (2.3), but |P L| is replaced by the shorter length |V S |; in the Pythagorean terminology, the square on the ordinate is equal to the rectangle with width equal to the abcissa applied to the segment V S , falling short of P L. The Greek for “falling short” is ἔλλειψιζ, and Apollonius calls γ an ellipse in this case. To obtain the rectangular equation of the ellipse, we set d = PP′ ...
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