CHAPTER VIII POLAR COÖRDINATES 100. Definition. Some topics in analytic geometry can be better investigated by the use of polar coördinates than by rectangular coördinates. In the polar system the position of a point is fixed by measuring a distance and a direction instead of by the measures of two distances. This is essentially the same system as that of bearing and distance used in surveying, contrasted with that of latitude and longitude used in geography. Choose a fixed point O as the origin, called the pole, and a fixed line OA through it, called the polar axis. Then any P(P,0) A + point P is determined if we know its distance from O and the angle that OP makes with OA. The measures of the distance OP and the angle AOP are called the polar coördinates of P and are designated by p and 6. The distance ρ is called the radius vector of P, and 0 is called the vectorial angle. Polar coördinates do not obey the conventions of the rectangular system as to their direction and magnitude. As in trigonometry the radius vector may be rotated indefinitely in a counter-clockwise or clockwise direction, making take on any positive or negative value. Distances measured on the terminal line of away from the pole are positive; those measured in the opposite direction, on the terminal produced, are negative. (See the right-hand diagram on page 127.) Hence every pair of real numbers (p, 0) determines one point which may be located according to the following rule. RULE FOR PLOTTING. Taking the polar axis as an initial line, lay off the vectorial angle 0, counter-clockwise if positive, clockwise if negative. Then measure off the radius vector p, on the terminal of 0 if positive, on the terminal of 0 produced through the pole if negative. Since and +2 have the same terminal line, a point may be represented by an indefinite number of pairs of 101. Relations between Rectangular and Polar Coördinates. -Take the pole at the origin of rectangular coördinates and the polar axis as the positive half of the x-axis. From the figures and the definitions of the trigonometric functions it is evident that for P(x, y) = P(p, 0) in any quadrant the following formulas are true: These equations enable us to transform rectangular equations and coördinates into polar forms, and conversely. NOTE. In transforming the rectangular coördinates of a point into polar coördinates, care should be taken to group together the corresponding values of p and 0. 102. Polar Curves. The definitions of equation and locus in polar are the same as those in rectangular coördinates, if (p, 0) is substituted for (x, y) (§ 14). The equation in polar coördinates is derived as in the rectangular system (§ 16). In a few cases the polar equation may be obtained best by deriving it in the rectangular form and then substituting for x and y their values in terms of and and vice versa. ρ Plotting in polar coördinates resembles that in rectangular coördinates. The equation should usually be solved for ρ and a table of values formed, taking values of 0 at intervals of 30° (or sometimes 15°). When the curve has symmetry, it is usually unnecessary to carry the table through more than two quadrants. PROBLEMS 1. Plot the points: (a) (6, ± 30°); (b) (± 10, 420°); (c) (8, π to the pole, polar axis, and the 90° axis, and find their coördinates. 3. As in Problem 2 find the points symmetrical to: п (a) (16, 4TM); (b) (− 6, – 2*); (c) (− 10, – 3x). 2 4. Fix P(p, 0) in any quadrant and find its symmetry to: (a) P (p, 0); (b) P(- p, 0); (c) P(p, π 0). 5. Find the rectangular coördinates of the following points: 6. Find two pairs of polar coördinates for each of the following points and plot the point in each case : 7. Plot each of the following equations and identify each curve by transforming its equation to rectangular coördinates: 8. Draw each of the following curves and transform its equation 103. The Equation of the Straight Line. The general equation of the straight line in polar coördinates is not as convenient as the equation in rectangular coördinates and will not be discussed. The special cases where the line is parallel or perpendicular to the polar axis or passes through the pole lead to very simple equations. In the figure let the line 7 be perpendicular to the polar axis OA and have the polar intercept a. For any point P(p, 0) on the line it is evident that cos 0 the line is -a α whence the equation of Ρ Similarly the equation of a line parallel to the polar axis, of 90° intercept a, is For a line passing through the pole the equation is evidently Exercise 1. By transforming the normal form of the straight-line equation into polar coördinates show that the equation of any line in polar coördinates is p cos (0 — w) = p. As in the case of the 104. The Equation of the Circle. straight line the general form of the circle equation is not often used. Several special forms, which are common, are: Circle with center at pole, radius r: P Ρ = r; (38) = 2 r cos 0; (39) |