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+z Plane extends forever in all directions
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When we set the height equal to a constant in cylindrical coordinates, we get a plane parallel to the reference plane
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Some texts will tell you that the cylindrical coordinates of a point are listed in an ordered triple with the radius first, then the angle, and finally the height, as P = (r,q,h) Don t let this notational inconsistency baffle you For any particular set of coordinate values, we re talking about the same point, regardless of the order in which we list them In this book, we indicate the angle before the radius to be consistent with the polar-coordinate system described in Chap 3 When traveling from the origin out to some point P in space in the cylindrical system, most people find it easiest to think of the reference-plane angle q first (as in face northwest ), then the radius r (as in walk 40 meters ), and finally the height h (as in dig down 2 meters to find the treasure ) That s why, in this book, we use the form P = (q,r,h)
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What do we get if we set the direction angle q equal to a constant in cylindrical coordinates As an example, draw a diagram showing the graph of the equation q = p /2
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Alternative Three-Space
Solution
Let s think back again to Chap 3 In polar coordinates, if we set the direction angle equal to a constant, we get a line passing through the origin Cylindrical coordinates are simply a vertical extension of polar coordinates, going infinitely upward and infinitely downward If we hold q constant in cylindrical coordinates but allow the other coordinates to vary at will, we get a vertical plane, which is an infinite vertical extension of a horizontal line If k is any real-number constant, then the graph of q=k is a plane that passes through the vertical axis In the case where q = p /2, that plane also contains the ray for the direction angle p /2, as shown in Fig 9-4
+z Plane extends forever in all directions
Constant angle
q = p /2
Reference plane
Figure 9-4
When we set the angle equal to a constant in cylindrical coordinates, we get a plane that contains the vertical axis
Cylindrical Conversions
Conversion of coordinate values between cylindrical and Cartesian three-space is just as easy as conversion between polar and Cartesian two-space The only difference is that in threespace, we add the vertical dimension In xyz space, it s z; in cylindrical three-space, it s h
Cylindrical to Cartesian Let s look at the simplest conversions first These transformations are like going down a river; we can simply get into the boat (sharpen our pencils) and make sure we don t run aground
Cylindrical Conversions
(make an arithmetic error) Suppose we have a point (q,r,h) in cylindrical coordinates We can find the Cartesian x value of this point using the formula x = r cos q The Cartesian y value is y = r sin q The Cartesian z value is z=h
An example Consider the point (q,r,h) = (p,2, 3) in cylindrical coordinates Let s find the (x,y,z) representation in Cartesian three-space using the preceding formulas Plugging in the numbers gives us
x = 2 cos p = 2 ( 1) = 2 y = 2 sin p = 2 0 = 0 z = h = 3 Therefore, we have the Cartesian equivalent point (x,y,z) = ( 2,0, 3)
Cartesian to cylindrical: finding q Going from Cartesian to cylindrical coordinates is like navigating up a river We not only have to go against the current (do some hard work), but we have to be sure we take the right tributary (use the correct angle values) Cartesian-to-cylindrical angle conversion is the same as the Cartesian-to-polar angle conversion process that we learned in Chap 3 That was messy, because we had to break the situation down into nine different ranges for q In the cylindrical context, the angle-conversion process works as follows:
q=0 by default q=0 q = Arctan ( y /x) q = p /2 q = p + Arctan ( y /x) q=p q = p + Arctan ( y /x) q = 3p /2 q = 2p + Arctan ( y /x) When x = 0 and y = 0 that is, at the origin When x > 0 and y = 0 When x > 0 and y > 0 When x = 0 and y > 0 When x < 0 and y > 0 When x < 0 and y = 0 When x < 0 and y < 0 When x = 0 and y < 0 When x > 0 and y < 0
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