Engineering Calculus Notes 317

# Engineering Calculus Notes 317 - QP lies in the plane = ;...

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3.5. SURFACES AND THEIR TANGENT PLANES 305 We are interested, however, not in this circle, but in the surface consisting of points in R 3 at distance b from this circle, where 0 < b < a ; this is called a torus . It is reasonable to assume (and this will be veriFed later) that for any point P not on the circle, the nearest point to to P on the circle lies in the vertical plane given by Fxing θ at its value for P , say θ = α . This means that if P has cylindrical coordinates ( r,α,z ) then the nearest point to P on the circle is the point Q ( a cos α,a sin α, 0) as given above. The vector −−→
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Unformatted text preview: QP lies in the plane = ; its length is, by assumption, b , and if we denote the angle it makes with the radial line O Q by (igure 3.18 ), then we have x y z Q P a b Q P igure 3.18: Parametrization of Torus QP = ( b cos ) v + ( b sin ) k where v = (cos ) + (sin ) is the horizontal unit vector making angle with the x-axis. Since O Q = a v = ( a cos ) + ( a sin )...
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