Engineering Calculus Notes 157

Engineering Calculus Notes 157 - 145 2.2. PARAMETRIZED...

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Unformatted text preview: 145 2.2. PARAMETRIZED CURVES → while rotating − the same amount yields the vector making angle α (counterclockwise) with the positive y -axis, or equivalently making angle α + π with the positive x-axis: 2 → − = cos(α + π )− + sin(α + π )− → → u2 ı 2 2 → → = (− sin α)− + (cos α)− ı √ 1→ 3− →. =− − + ı 2 2 Our parametrization of the rotated ellipse is obtained from the standard → → → → parametrization by replacing − with − 1 and − with − 2 : ı u u → − (θ ) = (a cos θ )− + (b sin θ )− → → p u u 1 2 → → → → = (a cos θ ) (cos α)− + (sin α)− + (b sin θ ) (− sin α)− + (cos α)− ı ı → → = (a cos θ cos α − b sin θ sin α)− + (a cos θ sin α + b sin θ cos α)− ı √ √ a3 a b b3 → → = ı cos θ − sin θ − + − cos θ + sin θ − 2 2 2 2 or, in terms of coordinates, √ a3 b x= 2 cos θ − 2 sin θ √ y = − a cos θ + b 2 3 sin θ 2 Of course, we can combine these operations, but again some care is necessary: rotate the standard parametrization before adding the displacement; otherwise you will have rotated the displacement, as well. → For example, a parametrization of the ellipse centered at − = (1, 2) with c π axes rotated 6 radians counterclockwise from the positive coordinate axes is given (in terms of the notation above) by → − (θ ) = − + (a cos θ )− + (b sin θ )− → → → p c u1 u2 → → = (− + 2− ) + ı = √ a b3 → − cos θ + sin θ − 2 2 √ a b3 → 2 − cos θ + sin θ − 2 2 √ b a3 → ı cos θ − sin θ − + 2 2 √ a3 b → 1+ ı cos θ − sin θ − + 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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