LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and a half-line known as the pola axis, shown extending fom P to the ight in Figue 1 below. In pola coodinates, lines occu in two species. A line though the pole, making angle θ 0 with the pola axis, has an equation (1.1) θ = constant = θ 0. If a line l does not pass though the pole, the nomal line fom the pole to l intesects l at its point neaest the pole, say Q(d, θ 0 ) with d > 0. So the equation of l is (see Figue 1) (1.) = d/ cos(θ θ 0 ) = d sec(θ θ 0 ). Figue 1. Equation of a Line in Pola Coodinates Since d is the distance fom the pole P to the foot Q of the nomal, we call d the pedal 1 distance. 1 pedal: of o elating to the foot. 1
ROGER ALEXANDER 1.1. Pola-coodinate equation fo the line though given points. Given points A( 1, θ 1 ) and B(, θ ) we see that the line AB passes though the Pole if and only if θ θ 1 is an integal multiple of π; and then its equation is θ = θ 1. If AB does not pass though the Pole, we want to find paametes θ 0 and d so that the line though A and B is epesented by Equation (1.). We may suppose (1) adii 1 and have the same sign (if sgn( ) sgn( 1 ) eplace (, θ ) by (, θ +π); and () the angles satisfy θ θ 1 < π (if not, eplace θ by θ + kπ fo suitable intege k.) Figue. Paametes fo the Line Though Given Points. Figue shows that the pedal length d is the altitude on the base AB in P AB. It follows that Aea ( P AB) (1.3) d =. AB In P AB the altitude on the base P A (not shown in Figue ) is (1.4) h = sin(θ θ 1 ), so that, using (1.5) σ = sgn(sin(θ θ 1 )), we have (1.6) Aea ( P AB) = 1 sin(θ θ 1 ) = σ 1 sin(θ θ 1 ), since 1 and have the same sign.
LINES AND TANGENTS IN POLAR COORDINATES 3 By the Law of Cosines in P AB we have (1.7) AB = 1 + 1 cos(θ θ 1 ). Inseting Equations (1.6) and (1.7) into the fomula (1.3) fo d gives (1.8) d = σ 1 sin(θ θ 1 ) 1 + 1 cos(θ θ 1 ) It is convenient to expess θ 0 in the fom (see Figue ) (1.9) θ 0 = θ 1 + ϕ. Then the equation 1 cos(θ 1 θ 0 ) = d becomes (1.10) cos ϕ = d 1 = AB σ sin(θ θ 1 ). To find sin ϕ we expand the equation cos (θ (θ 1 + ϕ)) = d and simplify, leading to (1.11) sin ϕ = σ AB ( cos(θ θ 1 ) 1 ). Finally, using Equation (1.9) and the addition fomulas fo sine and cosine, we eadily find that (1.1) cos θ 0 = σ( sin θ 1 sin θ 1 ) 1 + 1 cos(θ θ 1 ) (1.13) sin θ 0 = σ( cos θ 1 cos θ 1 ) 1 + 1 cos(θ θ 1 ). Execises fo Section 1 1.1 Fo each pai of points given in pola coodinates, find a polacoodinate equation fo the line detemined by the points. (a) ( 1, θ 1 ) = ( 3, 0), (, θ ) = (1, π/). (b) ( 1, θ 1 ) = (6, π/), (, θ ) = (3, 3π/4). (c) ( 1, θ 1 ) = (4, π/), (, θ ) = (8, 3π/). (d) ( 1, θ 1 ) = (5, 0), (, θ ) = (5, π/4). 1. (a) Explain why evey line in the x-y plane that does not pass though the oigin has an equation of the fom ax + by = c with c > 0, and a, b not both zeo. (Use the fact that any line not passing though the oigin eithe is vetical o has a slope-intecept equation with a nonzeo intecept.)
4 ROGER ALEXANDER (b) Substitute x = cos θ, y = sin θ into the equation fo the line and solve fo. Then show that if θ 0 and d ae defined by cos θ 0 = a a + b, sin θ 0 = b a + b, d = c a + b, the equation fo the line takes the standad fom (1.). 1.3 The Catesian coodinates of A and B in Figue ae (x i, y i ) = ( i cos θ i, i sin θ i ), i = 1,. Show that if the angle α is defined by cos α = x x 1 AB, sin α = y y 1 AB, then the angle θ 0 = θ 1 + ϕ satisfies θ 0 = α π/. 3. Pola Equation fo the Tangent Line Suppose that a pola cuve is defined by = f(θ) with a continuously diffeentiable function f defined on some open θ-inteval, and that θ 1 is an inteio point of this inteval. Set 1 = f(θ 1 ); we seek the equation of the tangent line to the cuve at ( 1, θ 1 ). What the equation fo the tangent line is depends on whethe 1 = 0. We conside fist the case 1 0 and then the case 1 = 0. 3.1. Tangent at a point whee 1 0. Let θ be an incement and wite θ = θ 1 + θ, and = f(θ ). Since f is continuous, 1 and will have the same sign fo all sufficiently small incements θ. We find the paametes θ 0, d fo the tangent line as limits of the coesponding paametes fo chods joining A( 1, θ 1 ) and B(, θ ) as θ 0; see Figue 3. Since f is diffeentiable thee ae (see the Appendix, Section 5) wobble functions w f, w c and w s of θ, all with limit 0 as θ 0, so that (witing 1 = f (θ 1 )) (3.1) (3.) (3.3) f(θ 1 + θ) = 1 + θ ( 1 + w f ( θ) ), cos(θ 1 + θ) = cos θ 1 + θ ( sin θ 1 + w c ( θ)), sin(θ 1 + θ) = sin θ 1 + θ (cos θ 1 + w s ( θ)). Fist we detemine the limit of the pedal distance d. In ou notation the quantity inside the squae oot in the denominato in Equation (1.8) is + 1 1 + 1 (1 cos θ) = ( 1 ) + 4 1 sin 1 θ (3.4) = ( θ) ( 1 + w f ) + 4 1 sin 1 θ.
LINES AND TANGENTS IN POLAR COORDINATES 5 Figue 3. Chod ABC and Tangent AT. Taking d fom Equation (1.8), dividing numeato and denominato by θ and taking the limit θ 0 we find (using σ θ = θ = ( θ) ) lim d = lim θ 0 θ 0 1 sin( θ) θ ( 1 + w f ) + 4 1 sin 1 θ ( θ) (3.5) = 1 1 + ( 1 ). To find the angle θ 0 fo the tangent line, epesent θ 0 = θ 1 + ϕ as we did in Section 1. We calculate the limit of cos ϕ fom Equation (1.10) to be lim cos ϕ = lim θ 0 θ 0 σ sin( θ). AB
6 ROGER ALEXANDER Divide numeato and denominato by θ and use Equation (3.4) to get (3.6) (3.7) lim cos ϕ = lim θ 0 θ 0 = sin( θ) θ 1 σ θ AB 1 1 + ( 1 ). A simila calculation fo the limit of sin ϕ (Equation (1.11)) leads to σ( cos(θ θ 1 ) 1 ) lim sin ϕ = lim θ 0 θ 0 AB σ = lim θ 0 (( 1 ) + ( 1 cos(θ θ 1 ) )) AB and, dividing numeato and denominato by σ( θ),, (3.8) 1 = lim θ θ 0 = 1 1 + ( 1 ). sin 1 θ θ 1 σ θ AB Combining the equations fo d, cos ϕ and sin ϕ we find that the paametes of the tangent line at a point ( 1, θ 1 ) with 1 0 ae simply expessed by (3.9) d = 1 cos ϕ, θ 0 = θ 1 + ϕ. 3.. Smooth cuves; ac length in pola coodinates. A fee dividend of ou deivation is the fomula fo the diffeential of ac length in pola coodinates. The denominato in the fomula (1.8) fo d is just the length of the chod fom ( 1, θ 1 ) to (, θ ), o s. Hence the quantity unde the squae oot in the denominato of (3.5) gives ds, the squaed diffeential of ac length: (3.10) ds = + ( ) dθ. Recall that a cuve is smooth if the diffeential fom ds neve vanishes. We see that the condition fo a pola-coodinate cuve to be smooth is that and ae neve simultaneously zeo. 3.3. Tangent at the pole. Suppose now that a smooth cuve given by = f(θ) passes though the pole at a cetain paamete value θ = θ 1 : f(θ 1 ) = 0, f (θ 1 ) 0. When θ is small, the point B(f(θ 1 + θ), θ 1 + θ) is given by ( ) ( ) f(θ 1 ) + (f (θ 1 ) + w f ) θ, θ 1 + θ = (f (θ 1 ) + w f ) θ, θ 1 + θ
LINES AND TANGENTS IN POLAR COORDINATES 7 and the chod P B has the diection θ 1 + θ. In the limit θ 0 the diection is obviously θ 1. The tangent line to a smooth cuve at a point whee f(θ 1 ) = 0 is the line θ = θ 1. 4. Execises fo Section 3 3.1 A cicle with diamete a lying along the pola axis with one end at the pole has the equation = a cos θ, 0 θ π. Show that the paametes ϕ and d of the tangent line at a point (, θ) on the cicle ae given by ϕ = θ, d = a cos θ. Find all values of θ whee the tangent line is (a) hoizontal; (b) vetical. a 3. Fo the paabola =, show that the paametes ϕ and d 1 + cos θ of the tangent line at a point (, θ) ae given by ϕ = 1 θ, d = a sec 1 θ. (Use 1 + cos θ = cos 1 θ.) 3.3 Fo the cadioid = a(1 + cos θ), show that the paametes ϕ and d of the tangent line at a point (, θ) ae given by ϕ = 1 θ, d = a cos3 1 θ. 5. Appendix: Diffeentiability and the Wobble Function Suppose a function f is defined on some open inteval containing a point a, and is diffeentiable at a. Then (5.1) f f(a + h) f(a) (a) = lim. h 0 h Subtacting the constant f (a) fom both sides and eaanging, we get ( ) f(a + h) f(a) (5.) lim f (a) = 0. h 0 h We define the wobble function w(h) to be the paenthesized expession on the left hand side. We call it the wobble function because w(h) is the diffeence between slopes of the chod and the tangent line. Now take the equation defining the wobble function, f(a + h) f(a) w(h) = f (a), h and solve fo f(a + h). The esult is (5.3) f(a + h) = f(a) + ( f (a) + w(h) ) h. This simple calculation shows that if f is diffeentiable at a, then thee is a wobble function w defined fo small h so that Equation (5.3) holds. (The convese is also tue, but it is not needed hee.)