Engineering Calculus Notes 165

Engineering Calculus Notes 165 - x = cos 2 z y = sin 2 z...

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2.2. PARAMETRIZED CURVES 153 which describes a point whose projection on the xy -plane moves counterclockwise in a circle of radius 1 about the origin; as this projection circulates around the circle, the point itself rises in such a way that during a complete “turn” around the circle, the “rise” is one unit. The “corkscrew” curve traced out by this motion is called a helix (Figure 2.22 ). x y z Figure 2.22: Helix While this can be considered as the locus of the pair of equations
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Unformatted text preview: x = cos 2 z y = sin 2 z such a description gives us far less insight into the curve than the parametrized version. As another example, let us parametrize the locus of the pair of equations x 2 + y 2 = 1 y + z = 0 which, geometrically, is the intersection of the vertical cylinder x 2 + y 2 = 1 with the plane y + z = 0 . The projection of the cylinder on the xy-plane is easily parametrized by x = cos t y = sin t...
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