Engineering Calculus Notes 317

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 −−→
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 )...
View Full Document

Ask a homework question - tutors are online