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Engineering Calculus Notes 162

Engineering Calculus Notes 162 - (Figure 2.19 It is...

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150 CHAPTER 2. CURVES position from which it started, and if, at the same time as the straight line revolves, a point move at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane. Of course, Archimedes is describing the above curve for the variation of θ from 0 to 2 π . If we continue to increase θ beyond 2 π , the curve continues to spiral out, as illustrated in Figure 2.18 . If we include negative values of Figure 2.18: The Spiral of Archimedes, r = θ , θ 0 θ , we get another spiral, going clockwise instead of counterclockwise
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Unformatted text preview: (Figure 2.19 ) It is di±cult to see how to write down an equation in x and y Figure 2.19: r = θ , θ < with this locus. Finally, we consider the cycloid , which can be described as the path of a point on the rim of a wheel rolling along a line (Figure 2.20 ). Let R be the radius of the wheel, and assume that at the beginning the point is located on the line—which we take to be the ξ x —at the origin, so the center of the...
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