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Imagine yourself at some point far from the origin on the +x axis in Fig 18-14, or far from the origin on the +y axis in Fig 18-15, or far from the origin on the +z axis in Fig 18-16 If you have excellent spatial perception, you ll be able to figure out that in all three of these situations, the constant c is negative! In each case, a retreating point on the helix will appear to revolve counterclockwise, and an approaching point on the helix will appear to revolve clockwise
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Here s a challenge!
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Suppose that we encounter an object in Cartesian xyz space whose parametric equations are x = 2 cos t y = 3 sin t z = 3t What sort of object is this
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Let s divide the first two equations through by their respective constants That gives us x /2 = cos t and y /3 = sin t Squaring both sides of both equations, we obtain (x /2)2 = cos2 t
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Circular Helixes
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and (y /3)2 = sin2 t When we add these two equations, left-to-left and right-to-right, we get (x /2)2 + (y /3)2 = cos2 t + sin2 t The rules of trigonometry tell us that cos2 t + sin2 t = 1 so the preceding equation can be rewritten as (x /2)2 + (y /3)2 = 1 and further morphed into x2/4 + y2/9 = 1 This equation describes an ellipse in the xy plane whose horizontal (x-coordinate) semi-axis measures 2 units, and whose vertical (y-coordinate) semi-axis measures 3 units Now let s consider the z coordinate The equation for z in terms of t is z = 3t This equation tells us that a point on our object travels in the negative z direction as the value of the parameter t increases The complete set of three parametric equations therefore describes an elliptical helix centered on the z axis As we move in the positive z direction, the helix rotates clockwise, because the coefficient of t is negative It looks something like the helix in Fig 18-16, except that it s stretched by approximately 50 percent in the positive and negative y directions (vertically in this particular illustration)
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Here s an extra-credit challenge!
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Sketch three-dimensional perspective graphs of the helixes described in the foregoing two examples and challenge
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You re on your own That s why you get extra credit!
Lines and Curves in Three-Space
Practice Exercises
This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Consider the following three-way equation for a straight line in Cartesian xyz space: x 1=y 2=z 4 Find a point on the line, find the preferred direction numbers, and determine the direction vector as a sum of multiples of i, j, and k 2 Consider the following three-way equation for a straight line in Cartesian xyz space: 4x = 5y = 6z Find a point on the line, find the preferred direction numbers, and determine the direction vector as a sum of multiples of i, j, and k 3 Consider the following three-way equation for a straight line in Cartesian xyz space: (x 2)/3 = (4y 8)/4 = (z + 5)/( 2) Find a point on the line, find the preferred direction numbers, and determine the direction vector as a sum of multiples of i, j, and k 4 Consider a relation in Cartesian xyz space described by the system of parametric equations x = 4 y=t z = t2 1 Draw a two-dimensional graph of this relation as it appears when we look broadside at the plane containing it 5 Consider a relation in Cartesian xyz space described by the system of parametric equations x = t2 + 2t y=t z=0
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