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Unformatted text preview: MATH 118, LECTURE 12: POLAR COORDINATES 1 Polar Coordinates For many applications, the dependence of y on x ( y = f ( x )) is not the most practical approach to modelling a problem. For instance, imagine you are manning a lighthouse late at night when you see a ship on the horizon (see Figure 1). You are interested in ascertaining the position of the boat based on your observations, but are unsure how to go about it. Since you are observing the boat from a fixed position, it is not practical to consider the boat’s position as a Cartesian grid ( x, y )—at least not at first—since this requires first obtaining an x coordinate, and you are not permitted to move. You would much rather have a frame of reference which allows you to remain stationary. The question then becomes whether we can obtain enough information to accurately describe the position of the boat, so that we could pass the information on to somebody else and they would be able to know exactly where the boat is (in whichever frame of reference they favour). x y direction d i s t a n c e Figure 1: Measuring the position of a boat from a stationary point of refer- ence. 1 The answer, of course, is yes. Even though we are stationary, we can still quantify the boat’s direction (e.g. NNW, SE, etc.) and distance (e.g. 1.2 km, etc.) relative to our current position. We could pass this information on to a third party and, so long as they knew our precise position (our frame of reference), they would be able to know precisely the position of the boat. They could also use this information (direction and distance) to construct a Cartesian grid (longitude and latitude) giving the boat’s position....
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This note was uploaded on 04/14/2009 for the course ENGINEERIN MATH 118 taught by Professor Soares during the Spring '09 term at Waterloo.
- Spring '09