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Let s look at the graphs of three generalized equations in polar coordinates In Cartesian coordinates, all equations of these forms produce straight-line graphs Only one of them does it now!
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Constant angle When we set the direction angle to a numerical constant, we get a simple polar equation of the form
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q=a where a is the constant As we allow the value of r to range over all the real numbers, the graph of any such equation is a straight line passing through the origin, subtending an angle of a with respect to the reference axis Figure 3-2 shows two examples In these cases, the equations are q = p /3 and q = 7p /8
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p /2
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q = p /3
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q = 7p/8
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3p/2
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Figure 3-2 When we set the angle constant, the graph is a
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straight line through the origin Here are two examples
Three Basic Graphs
Constant radius Imagine what happens if we set the radius to a numerical constant This gives us a polar equation of the form
r=a where a is the constant The graph is a circle centered at the origin whose radius is a, as shown in Fig 3-3, when we allow the direction angle q to rotate through at least one full turn of 2p If we allow the angle to span the entire set of real numbers, we trace around the circle infinitely many times, but that doesn t change the appearance of the graph
Angle equals radius times positive constant Now let s investigate a more interesting situation Figure 3-4 shows an example of what happens in polar coordinates when we set the radius equal to a positive constant multiple of the angle We get a pair of mirror-image spirals To see how this graph arises, imagine a ray pointing from the origin straight out toward the right along the reference axis (labeled 0) The angle is 0, so the radius is 0 Now suppose the ray starts to rotate counterclockwise, like the sweep on an old-fashioned military radar screen The angle increases positively at a constant rate Therefore, the radius also increases at a constant rate, because the radius is a positive constant multiple of the angle The resulting
p/ 2
3p /2
Figure 3-3 When we set the radius constant, the
graph is a circle centered at the origin In this case, the radius is an arbitrary value a
Polar Two-Space
p /2
3p /2
Figure 3-4 When we set the radius equal to a positive
constant multiple of the angle, we get a pair of spirals
graph is the solid spiral The pitch (or tightness ) of the spiral depends on the value of the constant a in the equation r = aq Small positive values of a produce tightly curled-up spirals Larger positive values of a produce more loosely pitched spirals Now suppose that the ray starts from the reference axis and rotates clockwise At first, the angle is 0, so the radius is 0 As the ray turns, the angle increases negatively at a constant rate That means the radius increases negatively at a constant rate, too, because we re multiplying the angle by a positive constant We must plot the points in the exact opposite direction from the way the ray points When we do that, we get the dashed spiral in Fig 3-4 The pitch is the same as that of the heavy spiral, because we haven t changed the value of a The entire graph of the equation consists of both spirals together
Angle equals radius times negative constant Figure 3-5 shows an example of what happens in polar coordinates when we set the radius equal to a negative constant multiple of the angle As in the previous case, we get a pair of spirals, but they re upside-down with respect to the case when the constant is positive To see
Three Basic Graphs