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Figure 9-1 is a functional diagram of a system of cylindrical coordinates It s basically a polar coordinate plane of the sort we learned about in Chap 3, with the addition of a height axis to define the third dimension
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How it works To set up a cylindrical coordinate system, we paste a polar plane onto a Cartesian xy plane, creating a reference plane We call the positive Cartesian x axis the reference axis Imagine a point P in three-space, along with its projection point P onto the reference plane In this context, the term projection means that P is directly above or below P, so a line connecting the two points is perpendicular to the reference plane We define three coordinates:
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The direction angle, which we call q, is the angle in the reference plane as we turn counterclockwise from the reference axis to the ray that goes out from the origin through P The radius, which we call r, is the straight-line distance from the origin to P The height, which we call h, is the vertical displacement (positive, negative, or zero) from P to P These three coordinates give us enough information to uniquely define the position of P as shown in Fig 9-1 We express the cylindrical coordinates as an ordered triple P = (q,r,h)
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Figure 9-1 Cylindrical coordinates define points in
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three dimensions according to an angle, a radial distance, and a vertical displacement
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Strange values We can have nonstandard direction angles in cylindrical coordinates, but it s best to add or subtract whatever multiple of 2p to bring the angle into the preferred range of 0 q < 2p If q 2p, then we re making at least one complete counterclockwise rotation from the reference axis If q < 0, then we re rotating clockwise from the reference axis rather than counterclockwise We can have negative radii, but it s best to reverse the direction angle if necessary to keep the radius nonnegative We can multiply a negative radius coordinate by 1 so it becomes positive, and then add or subtract p to or from the direction angle to ensure that 0 q < 2p The height h can be any real number We have h > 0 if and only if P is above the reference plane, h < 0 if and only if P is below the reference plane, and h = 0 if and only if P is in the reference plane An example In the situation shown by Fig 9-1, the direction angle q appears to be somewhat more than p (half of a rotation from the reference axis) but less than 3p /2 (three-quarters of a rotation) The radius r is positive, but we can t tell how large it is because there are no coordinate increments for reference The height h is also positive, but again, we don t know its exact value because there are no reference increments
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+z Cylinder extends upward forever r=k Constant radius
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Cylinder extends downward forever
Reference plane
Figure 9-2
When we set the radius equal to a constant in cylindrical coordinates, we get an infinitely tall vertical cylinder whose axis corresponds to the vertical axis
Another example In Chap 3, we learned that the equation of a circle in polar two-space is simple; all we have to do is specify a radius If we do the same thing in cylindrical three-space, we get a vertical cylinder that s infinitely tall, with an axis that corresponds to the vertical coordinate axis Figure 9-2 shows what we get when we graph the equation
r=k in cylindrical three-space, where k is a nonzero constant
Still another example If we set the height equal to a nonzero constant in cylindrical coordinates, we get the set of all points at a specific distance either above or below the reference plane That s always a plane parallel to the reference plane Figure 9-3 is an example of the generic situation where
h=k In this case, k is a positive real-number constant, but we don t know the exact value because the graph doesn t show us any reference increments for the height coordinate