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vb.net generate qr code Cylindrical Coordinates in VS .NET
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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 realnumber 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 94 +z Plane extends forever in all directions
Constant angle
q = p /2 Reference plane
Figure 94 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 threespace is just as easy as conversion between polar and Cartesian twospace The only difference is that in threespace, we add the vertical dimension In xyz space, it s z; in cylindrical threespace, 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 threespace 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) Cartesiantocylindrical angle conversion is the same as the Cartesiantopolar 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 angleconversion 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

