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Polar Two-Space
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Figure 3-1 The polar coordinate plane Angular
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divisions are straight lines passing through the origin Each angular division represents p/6 units Radial divisions are circles
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Strange values In polar coordinates, it s okay to have nonstandard direction angles If q 2p, it represents at least one complete counterclockwise rotation from the reference axis If the direction angle is q < 0, it represents clockwise rotation from the reference axis rather than counterclockwise rotation We can also have negative radius coordinates If we encounter some point for which we re told that r < 0, we can multiply r by 1 so it becomes positive, and then add or subtract p to or from the direction That s like saying Proceed 10 km due east instead of Proceed 10 km due west Which variable is which If you read a lot of mathematics texts and papers, you ll sometimes see ordered pairs for polar coordinates with the radius listed first, and then the angle Instead of the form (q,r), the ordered pairs will take the form (r,q) In this scheme, the radius is the independent variable, and the direction is the dependent variable It works fine, but it s easier for most people to imagine that the radius depends on the direction Think of an old-fashioned radar display like the ones shown in war movies made in the middle of the last century A bright radial ray rotates around a circular screen, revealing targets at various distances The rotation continues at a steady rate; it s independent Target distances are functions of the direction Theoretically, a radar display could work in the opposite sense with an expanding bright circle instead of a rotating ray, and all of the targets would show up
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in the same places But that geometry wasn t technologically practical when radar sets were first designed, and it was never used Let s use the (q,r) format for ordered pairs, where q is the independent variable and r is the dependent variable
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You ask, How we can write down relations and functions intended for polar coordinates as opposed to those meant for Cartesian coordinates It s simple When we want to denote a relation or function (call it f ) in polar coordinates where the independent variable is q and the dependent variable is r, we write r = f (q) We can read this out loud as r equals f of q When we want to denote a relation or function (call it g) in Cartesian coordinates where the independent variable is x and the dependent variable is y, we can write y = g (x) We can read this out loud as y equals g of x
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Here s a challenge!
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Provide an example of a graphical object that represents a function in polar coordinates when q is the independent variable, but not in Cartesian xy coordinates when x is the independent variable
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Solution
Consider a polar function that maps all inputs into the same output, such as f (q) = 3 Because f (q) is another way of denoting r, this function tells us that r = 3 The graph is a circle with a radius of 3 units In Cartesian coordinates, the equation of the circle with radius of 3 units is x2 + y2 = 9 (Note that 9 = 32, the square of the radius) If we let y be the dependent variable and x be the independent variable, we can rearrange this equation to get y = (9 x 2)1/2 We can t claim that y = g (x) where g is a function of x in this case There are values of x (the independent variable) that produce two values of y (the dependent variable) For example, if x = 0, then y = 3 If we want to say that g is a relation, that s okay; but g is not a function
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