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10 Polar Coordinates, Parametric Equations Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian ) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems. A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangle. In polar coordinates a point in the plane is identified by a pair of numbers ( r, θ ). The number θ measures the angle between the positive x -axis and a ray that goes through the point, as shown in figure 10.1; the number r measures the distance from the origin to the point. Figure 10.1 shows the point with rectangular coordinates (1 , 3) and polar coordinates (2 , π/ 3), 2 units from the origin and π/ 3 radians from the positive x -axis. 3 1 . . (2 , π/ 3) . Figure 10.1 Polar coordinates of the point (1 , 3). 217 218 Chapter 10 Polar Coordinates, Parametric Equations Just as we describe curves in the plane using equations involving x and y , so can we describe curves using equations involving r and θ . Most common are equations of the form r = f ( θ ). EXAMPLE 10.1 Graph the curve given by r = 2. All points with r = 2 are at distance 2 from the origin, so r = 2 describes the circle of radius 2 with center at the origin. EXAMPLE 10.2 Graph the curve given by r = 1 + cos θ . We first consider y = 1 + cos x , as in figure 10.2. As θ goes through the values in [0 , 2 π ], the value of r tracks the value of y , forming the “cardioid” shape of figure 10.2. For example, when θ = π/ 2, r = 1 + cos( π/ 2) = 1, so we graph the point at distance 1 from the origin along the positive y -axis, which is at an angle of π/ 2 from the positive x -axis. When θ = 7 π/ 4, r = 1 + cos(7 π/ 4) = 1 + 2 / 2 1 . 71, and the corresponding point appears in the fourth quadrant. This illustrates one of the potential benefits of using polar coordinates: the equation for this curve in rectangular coordinates would be quite complicated. 0 1 2 π/ 2 π 3 π/ 2 2 π . . . . . . . . . . . Figure 10.2 A cardioid: y = 1 + cos x on the left, r = 1 + cos θ on the right. Each point in the plane is associated with exactly one pair of numbers in the rect- angular coordinate system; each point is associated with an infinite number of pairs in polar coordinates. In the cardioid example, we considered only the range 0 θ 2 π , and already there was a duplicate: (2 , 0) and (2 , 2 π ) are the same point. Indeed, every value of θ outside the interval [0 , 2 π ) duplicates a point on the curve r = 1+cos θ when 0 θ < 2 π . We can even make sense of polar coordintes like ( - 2 , π/ 4): go to the direction π/ 4 and then move a distance 2 in the opposite direction; see figure 10.3. As usual, a negative angle θ means an angle measured clockwise from the positive x -axis. The point in figure 10.3 also has coordinates (2 , 5 π/ 4) and (2 , - 3 π/ 4).

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