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We can have nonstandard vertical angles, although things are simplest if we keep them nonnegative but no larger than p Theoretically, all possible locations in space can be covered if we restrict the vertical angle to the range 0 f p If it s outside this range, such as p < f < 0 or p < f < 2p, we can multiply the radius r by 1, and then add or subtract p to or from f, and we ll end up at the point we want But those are confusing ways to get there! The radius r can be any real number, but things are simplest if we keep it nonnegative If our horizontal and vertical direction angles put us on a ray that goes from the origin through P, then r > 0 If our direction angles put us on a ray that goes from the origin away from P, then r < 0 We have r = 0 if and only if P is at the origin If we find ourselves working with a negative radius, we should reverse the direction by adding or subtracting p to or from both angles, keeping 0 q < 2p and 0 f p Then we can take the absolute value of the negative radius and use it as the radius coordinate
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An example In the situation of Fig 9-5, the horizontal direction angle q appears to be somewhere between p and 3p /2 The vertical direction angle f appears to be roughly 1 radian We can t be sure of the exact values of these angles, because we don t have any reference lines to compare them with The radius r is positive, but we have no idea how large it is because there are no radial coordinate increments Another example Imagine that we set the horizontal direction angle q equal to a constant in spherical coordinates For example, let s say that we have the equation
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q = 7p /5 When we work in polar coordinates and set the direction angle equal to a constant, we get a line passing through the origin In spherical coordinates, the horizontal angle in the reference plane is geometrically identical to the polar direction angle Therefore, if k is any real-number constant, the graph of q=k is a plane that passes through the vertical axis When k = 7p /5, that vertical plane also contains the ray for the direction angle 7p /5, as shown in Fig 9-6
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Still another example If we set the radius equal to a constant in spherical coordinates, we get the set of all points at some fixed distance from the origin That s a sphere centered at the origin Figure 9-7 shows what happens when we graph the following equation:
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r=k in spherical three-space, where k is a nonzero constant
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Alternative Three-Space
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Figure 9-6
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When we set the horizontal angle equal to a constant in spherical coordinates, we get a plane that contains the vertical axis
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Constant radius
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Reference plane
Figure 9-7
When we set the radius equal to a constant in spherical coordinates, we get a sphere centered at the origin
Spherical Coordinates
Are you confused
Don t get the wrong idea about the meaning of the radius in spherical coordinates It s not the same as the cylindrical-coordinate radius! In spherical coordinates, the radius follows a straightline path from the origin to the point whose coordinates we re interested in This line almost never lies in the reference plane In cylindrical coordinates, the radius goes from the origin to the projection of the point in the reference plane You can see the difference if you compare Fig 9-1 with Fig 9-5
Are you still confused
If you ve read a lot of other pre-calculus texts (and I recommend that you do), you might notice that the order in which we list spherical coordinates is different from the way it s done in some of those other texts You might see the spherical coordinates of a point P go with the radius first, then the horizontal angle, and finally the vertical angle, as P = (r,q,f) Theoretically, it doesn t matter in which order we list the coordinates For any particular values, we re always working with the same point When we want to get from the origin to a point in spherical three-space, most people find it easiest to think of the horizontal angle q first (as in face southeast ), then the vertical angle f (as in fix your gaze at an angle that s p /6 radian from the zenith ), and finally the radius r (as in follow the string for 150 meters to reach the kite ) That s why we use the form P = (q,f,r)
Here s a challenge!
What sort of graph do we get if we set the vertical angle f equal to a constant in spherical coordinates As an example, draw a diagram showing the graph of the following equation: f = p /4
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