Engineering Calculus Notes 156

# Engineering - 144 CHAPTER 2 CURVES or equivalently x = a cos θ y = b sin θ Suppose we want to describe instead the ellipse with the same

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Unformatted text preview: 144 CHAPTER 2. CURVES or equivalently x = a cos θ . y = b sin θ Suppose we want to describe instead the ellipse with the same semi-axes (still parallel to the coordinate axes) but with center at the point (c1 , c2 ). The displacement vector taking the origin to this position is simply the position vector of the new center → − =c − +c − → → c 1ı 2 so we can obtain the new ellipse from the old simply by adding this (constant) vector to our parametrization function: − (θ ) = (c − + c − ) + (a cos θ − + b sin θ − ) → → → → → p ı 1ı 2 → − + (c + b sin θ )− → = (c + a cos θ ) ı 1 2 or, in terms of coordinates, x = c1 +a cos θ . y = c2 +b sin θ We might also consider the possibility of a conic section obtained by rotating a standard one. This is also easily accomplished for a → → parametrized expression: the role of − (resp. − ) is now played by a ı → − (resp. − ) of this vector. Two words of caution are in → rotated version u 1 u2 order here: the new vectors must still be unit vectors, and they must still be perpendicular to each other. Both of these properties are guaranteed if → → we make sure to rotate both − and − by the same amount, in the same ı direction. For example, suppose we want to describe the ellipse, still centered at the origin, with semi-axes a and b, but rotated counterclockwise from the → ı coordinate axes by α = π radians. Rotating − leads to the unit vector 6 π making angle α = 6 with the positive x-axis → − = (cos α)− + (sin α)− → → u1 ı √ 3− → + 1− → ı = 2 2 ...
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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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