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Look back at Fig 3-2 If you ponder this graph for awhile, you might suspect that the indicated equations aren t the only ones that can represent these lines You might ask, If we allow r to range over all the real numbers, both positive and negative, can t the line for q = p /3 also be represented by other equations such as q = 4p /3 or q = 2p /3 Can t the line representing the q = 7p /8 also be represented by q = 15p /8 or q = p /8 The answers to these questions are Yes When we see an equation of the form q = a representing a straight line through the origin in polar coordinates, we can add any integer multiple of p to the constant a, and we get another equation whose graph is the same line In more formal terms, a particular line q = a through the origin can be represented by q = kp a where k is any integer and a is a real-number constant
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What s the value of the constant, a, in the function shown by the graph of Fig 3-4 What s the equation of this pair of spirals Assume that each radial division represents 1 unit
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Note that if q = p, then r = 2 You can solve for a by substituting this number pair in the general equation for the pair of spirals Plugging in the numbers (q,r) = (p,2), proceed as follows: r = aq 2 = ap 2 /p = a Therefore, a = 2 /p, and the equation you seek is r = (2/p )q If you don t like parentheses, you can write it as r = 2q /p
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What is the polar equation of a straight line running through the origin and ascending at an angle of p /4 as you move to the right, with the restriction that 0 q < 2p If you drew this line on a standard Cartesian xy coordinate grid instead of the polar plane, what equation would it represent
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Two equations will work here They are q = p /4 and q = 5p /4 Keep in mind that the value of r can be any real number: positive, negative, or zero First, look at the situation where q = p /4 When r > 0, you get a ray in the p /4 direction When r < 0, you get a ray in the 5p /4 direction When r = 0, you get the origin point The union of these two rays and the origin point forms the line running through the origin and ascending at an angle of p /4 as you move toward the right Now examine events with the equation q = 5p /4 When r > 0, you get a ray in the 5p /4 direction When r < 0, you get a ray in the p /4 direction When r = 0, you get the origin point The union of the two rays and the origin point forms the same line as in the first case In the Cartesian xy plane, this line would be the graph of the equation y = x
Coordinate Transformations 45
Coordinate Transformations
We can convert the coordinates of any point from polar to Cartesian systems and vice versa Going from polar to Cartesian is easy, like floating down a river Getting from Cartesian to polar is more difficult, like rowing up the same river As you read along here, refer to Fig 3-6, which shows a point in the polar grid superimposed on the Cartesian grid
Polar to Cartesian Suppose we have a point (q,r) in polar coordinates We can convert this point to Cartesian coordinates (x,y) using the formulas