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Lines and Tangents in Polar Coordinates

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Lines and Tangents in Polar Coordinates LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polar-coordinate equations for lines A polar coordinate system in the plane is determined by a point P , called the pole, and a half-line known as the polar axis, shown extending from P to the right in Figure 1 below. In polar coordinates, lines occur in two species. A line through the pole, making angle θ0 with the polar axis, has an equation (1.1) θ = constant = θ0: If a line ` does not pass through the pole, the normal line from the pole to ` intersects ` at its point nearest the pole, say Q(d; θ0) with d > 0. So the equation of ` is (see Figure 1) (1.2) r = d= cos(θ − θ0) = d sec(θ − θ0): Figure 1. Equation of a Line in Polar Coordinates Since d is the distance from the pole P to the foot Q of the normal, we call d the pedal1 distance. 1pedal: of or relating to the foot. 1 2 ROGER ALEXANDER 1.1. Polar-coordinate equation for the line through given points. Given points A(r1; θ1) and B(r2; θ2) we see that the line AB passes through the Pole if and only if θ2 − θ1 is an integral multiple of π; and then its equation is θ = θ1. If AB does not pass through the Pole, we want to find parameters θ0 and d so that the line through A and B is represented by Equation (1.2). We may suppose (1) radii r1 and r2 have the same sign (if sgn(r2) 6= sgn(r1) replace (r2; θ2) by (−r2; θ2+π); and (2) the angles satisfy jθ2−θ1j < π (if not, replace θ2 by θ2 + 2kπ for suitable integer k.) Figure 2. Parameters for the Line Through Given Points. Figure 2 shows that the pedal length d is the altitude on the base AB in 4P AB. It follows that 2 · Area (4P AB) (1.3) d = : AB In 4P AB the altitude on the base PA (not shown in Figure 2) is (1.4) h = jr2jjsin(θ2 − θ1)j; so that, using (1.5) σ = sgn(sin(θ2 − θ1)); we have 2 · Area (4P AB) = jr1jjr2jjsin(θ2 − θ1)j (1.6) = σr1r2 sin(θ2 − θ1); since r1 and r2 have the same sign. LINES AND TANGENTS IN POLAR COORDINATES 3 By the Law of Cosines in 4P AB we have q 2 2 (1.7) AB = r1 + r2 − 2r1r2 cos(θ2 − θ1): Inserting Equations (1.6) and (1.7) into the formula (1.3) for d gives σr r sin(θ − θ ) (1.8) d = 1 2 2 1 p 2 2 r1 + r2 − 2r1r2 cos(θ2 − θ1) It is convenient to express θ0 in the form (see Figure 2) (1.9) θ0 = θ1 + ': Then the equation r1 cos(θ1 − θ0) = d becomes d r2 (1.10) cos ' = = σ sin(θ2 − θ1): r1 AB To find sin ' we expand the equation r2 cos (θ2 − (θ1 + ')) = d and simplify, leading to −σ (1.11) sin ' = (r2 cos(θ2 − θ1) − r1) : AB Finally, using Equation (1.9) and the addition formulas for sine and cosine, we readily find that σ(r sin θ − r sin θ ) (1.12) cos θ = 2 2 1 1 0 p 2 2 r1 + r2 − 2r1r2 cos(θ2 − θ1) −σ(r cos θ − r cos θ ) (1.13) sin θ = 2 2 1 1 0 p 2 2 r1 + r2 − 2r1r2 cos(θ2 − θ1) 2. Exercises for Section 1 1.1 For each pair of points given in polar coordinates, find a polar- coordinate equation for the line determined by the points. p (a) (r1; θ1) = ( 3; 0), (r2; θ2) = (1;p π=2). (b) (r1; θ1) = (6; π=2), (r2; θ2) = (3 2; 3π=4). (c) (r1; θ1) = (4; π=2), (r2; θ2) =p (8; 3π=2). (d) (r1; θ1) = (5; 0), (r2; θ2) = (5 2; π=4). 1.2 (a) Explain why every line in the x-y plane that does not pass through the origin has an equation of the form ax + by = c with c > 0, and a; b not both zero. (Use the fact that any line not passing through the origin either is vertical or has a slope-intercept equation with a nonzero intercept.) 4 ROGER ALEXANDER (b) Substitute x = r cos θ, y = r sin θ into the equation for the line and solve for r. Then show that if θ0 and d are defined by a b c cos θ0 = p ; sin θ0 = p ; d = p ; a2 + b2 a2 + b2 a2 + b2 the equation for the line takes the standard form (1.2). 1.3 The Cartesian coordinates of A and B in Figure 2 are (xi; yi) = (ri cos θi; ri sin θi); i = 1; 2: Show that if the angle α is defined by x − x y − y cos α = 2 1 ; sin α = 2 1 ; AB AB then the angle θ0 = θ1 + ' satisfies θ0 = α − π=2. 3. Polar Equation for the Tangent Line Suppose that a polar curve is defined by r = f(θ) with a continuously differentiable function f defined on some open θ-interval, and that θ1 is an interior point of this interval. Set r1 = f(θ1); we seek the equation of the tangent line to the curve at (r1; θ1). What the equation for the tangent line is depends on whether r1 = 0. We consider first the case r1 6= 0 and then the case r1 = 0. 3.1. Tangent at a point where r1 6= 0. Let ∆θ be an increment and write θ2 = θ1 + ∆θ, and r2 = f(θ2). Since f is continuous, r1 and r2 will have the same sign for all sufficiently small increments ∆θ. We find the parameters θ0; d for the tangent line as limits of the corresponding parameters for chords joining A(r1; θ1) and B(r2; θ2) as ∆θ ! 0; see Figure 3. Since f is differentiable there are (see the Appendix, Section 5) wobble functions wf , wc and ws of ∆θ, all with limit 0 as ∆θ ! 0, so that (writing 0 0 r1 = f (θ1)) 0 (3.1) f(θ1 + ∆θ) = r1 + ∆θ r1 + wf (∆θ) ; (3.2) cos(θ1 + ∆θ) = cos θ1 + ∆θ (− sin θ1 + wc(∆θ)) ; (3.3) sin(θ1 + ∆θ) = sin θ1 + ∆θ (cos θ1 + ws(∆θ)) : First we determine the limit of the pedal distance d. In our notation the quantity inside the square root in the denominator in Equation (1.8) is 2 2 2 2 1 r2 + r1 − 2r1r2 + 2r1r2(1 − cos ∆θ) = (r2 − r1) + 4r1r2 sin 2 ∆θ 2 0 2 2 1 (3.4) = (∆θ) (r1 + wf ) + 4r1r2 sin 2 ∆θ: LINES AND TANGENTS IN POLAR COORDINATES 5 Figure 3. Chord ABC and Tangent AT . Taking d from Equation (1.8), dividing numerator and denominator by ∆θ and taking the limit ∆θ ! 0 we find (using σ∆θ = j∆θj = p(∆θ)2) sin(∆θ) r1r2 lim d = lim ∆θ r 1 ∆θ!0 ∆θ!0 sin2 ∆θ 0 2 2 (r1 + wf ) + 4r1r2 (∆θ)2 r2 (3.5) = 1 : p 2 0 2 r1 + (r1) To find the angle θ0 for the tangent line, represent θ0 = θ1 + ' as we did in Section 1. We calculate the limit of cos ' from Equation (1.10) to be σr sin(∆θ) lim cos ' = lim 2 : ∆θ!0 ∆θ!0 AB 6 ROGER ALEXANDER Divide numerator and denominator by ∆θ and use Equation (3.4) to get sin(∆θ) r2 (3.6) lim cos ' = lim ∆θ ∆θ!0 ∆θ!0 1 σ∆θ AB r (3.7) = 1 : p 2 0 2 r1 + (r1) A similar calculation for the limit of sin ' (Equation (1.11)) leads to −σ(r cos(θ − θ ) − r ) lim sin ' = lim 2 2 1 1 ∆θ!0 ∆θ!0 AB σ (r1 − r2) + r2 1 − cos(θ2 − θ1) = lim ; ∆θ!0 AB and, dividing numerator and denominator by σ(∆θ), 2 1 r2 − r1 2 sin 2 ∆θ − − r2 = lim ∆θ ∆θ 1 ∆θ!0 AB σ∆θ −r0 (3.8) = 1 : p 2 0 2 r1 + (r1) Combining the equations for d, cos ' and sin ' we find that the parameters of the tangent line at a point (r1; θ1) with r1 6= 0 are simply expressed by (3.9) d = r1 cos '; θ0 = θ1 + ': 3.2. Smooth curves; arc length in polar coordinates. A free dividend of our derivation is the formula for the differential of arc length in polar coordinates. The denominator in the formula (1.8) for d is just the length of the chord from (r1; θ1) to (r2; θ2), or ∆s. Hence the quantity under the square root in the denominator of (3.5) gives ds2, the squared differential of arc length: p (3.10) ds = r2 + (r0)2 dθ: Recall that a curve is smooth if the differential form ds never vanishes. We see that the condition for a polar-coordinate curve to be smooth is that r and r0 are never simultaneously zero. 3.3. Tangent at the pole. Suppose now that a smooth curve given by r = f(θ) passes through the pole at a certain parameter value θ = θ1: 0 f(θ1) = 0; f (θ1) 6= 0: When ∆θ is small, the point B(f(θ1 + ∆θ); θ1 + ∆θ) is given by 0 0 f(θ1) + (f (θ1) + wf )∆θ; θ1 + ∆θ = (f (θ1) + wf )∆θ; θ1 + ∆θ LINES AND TANGENTS IN POLAR COORDINATES 7 and the chord PB has the direction θ1 + ∆θ. In the limit ∆θ ! 0 the direction is obviously θ1.
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