Polar Coordinates Graphing in Polar Coordinates

Lecture 12Section 9.3 Polar Coordinates Section 9.4 Graphing in Polar Coordinates 1 1.1 Polar Coordinates Polar Coordinates Polar Coordinate System The purpose of the polar coordinates is to represent curves that have symmetry about a point or spiral about a point. Frame of Reference In the polar coordinate system, the frame of reference is a point O that we call the pole and a ray that emanates from it that we call the polar axis. Polar Coordinates 1 Definition A point is given polar coordinates [r, θ] iff it lies at a distance |r| from the pole a long the ray θ, if r ≥ 0, and along the ray θ + π, if r < 0. Points in Polar Coordinates Points in Polar Coordinates • O = [0, θ] for all θ. • [r, θ] = [r, θ + 2nπ] for all integers n. • [r, −θ] = [r, θ + π]. 1.2 Relation to Rectangular Coordinates Relation to Rectangular Coordinates 2 Relation to Rectangular Coordinates • x = r cos θ, y = r sin θ. • r= p x2 + y 2 , θ = tan−1 ⇒ x2 + y 2 = r2 , tan θ = y x y . x Circles in Polar Coordinates Circles in Polar Coordinates In rectangular coordinates 2 In polar coordinates 2 2 r=a 2 2 r = 2a sin θ 2 2 r = 2a cos θ x +y =a 2 x + (y − a) = a 2 (x − a) + y = a x2 + y 2 = a2 ⇒ r2 = a2 x + (y − a) = a ⇒ x2 + y 2 = 2ay ⇒ 2 2 2 (x − a) + y = a ⇒ x2 + y 2 = 2ax ⇒ 2 2 2 Lines in Polar Coordinates 3 r2 = 2ar sin θ r2 = 2ar cos θ Lines in Polar Coordinates In rectangular coordinates In polar coordinates θ = α with α = tan−1 m y = mx x=a y=a r = a sec θ r = a csc θ y y = mx ⇒ = m ⇒ tan θ = m x x = a ⇒ r cos θ = a ⇒ r = a sec θ y = a ⇒ r sin θ = a ⇒ r = a csc θ 1.3 Symmetry Symmetry 4 Lemniscate (ribbon) r2 = cos 2θ cos[2(−θ)] = cos(−2θ) = cos 2θ [1ex]⇒ if [r, θ] ∈ graph, then [r, −θ] ∈ graph [1ex]⇒ symmetric about the x-axis. cos[2(π − θ)] = cos(2π − 2θ) = cos 2θ [1ex]⇒ if [r, θ] ∈ graph, then [r, π − θ] ∈ graph [1ex]⇒ symmetric about the y-axis. cos[2(π + θ)] = cos(2π + 2θ) = cos 2θ [1ex]⇒ if [r, θ] ∈ graph, then [r, π + θ] ∈ graph [1ex]⇒ symmetric about the origin. Lemniscates (Ribbons) r2 = a sin 2θ, r2 = a cos 2θ Lemniscate r2 = a sin 2θ sin[2(π + θ)] = sin(2π + 2θ) = sin 2θ [2ex]⇒ if [r, θ] ∈ graph, then [r, π + θ] ∈ graph [2ex]⇒ symmetric about the origin. 5 2 Graphing in Polar Coordinates 2.1 Spiral Spiral of Archimedes r = θ, θ ≥ 0 The curve is a nonending spiral. Here it is shown in detail from θ = 0 to θ = 2π. 2.2 Limaçons Limaçon (Snail): r = 1 − 2 cos θ • r = 0 at θ = 13 π, 53 π; |r| is a local maximum at θ = 0, π, 2π. • Sketch in 4 stages: [0, 13 π], [ 13 π, π], [π, 35 π], [ 53 π, 2π]. • cos(−θ) = cos θ ⇒ if [r, θ] ∈ graph, then [r, −θ] ∈ graph ⇒ symmetric about the x-axis. 6 Limaçons (Snails): r = a + b cos θ The general shape of the curve depends on the relative magnitudes of |a| and |b|. Cardioids (Heart-Shaped): r = 1 ± cos θ, r = 1 ± sin θ Each change cos θ → sin θ → − cos θ → − sin θ represents a counterclockwise rotation by 12 π radians. • Rotation by 12 π: r = 1 + cos(θ − 12 π) = 1 + sin θ. • Rotation by 12 π: r = 1 + sin(θ − 12 π) = 1 − cos θ. • Rotation by 12 π: r = 1 − cos(θ − 21 π) = 1 − sin θ. 2.3 Flowers Petal Curve: r = cos 2θ 7 • r = 0 at θ = π 3π 5π 7π 4, 4 , 4 , 4 ; |r| is a local maximum at θ = 0, π2 , π, 3π 2 , 2π. • Sketch the curve in 8 stages. • cos[2(−θ)] = cos 2θ, cos[2(π ± θ)] = cos 2θ ⇒ symmetric about the x-axis, the y-axis, and the origin. Petal Curves: r = a cos nθ, r = a sin nθ • If n is odd, there are n petals. • If n is even, there are 2n petals. 2.4 Intersections Intersections: r = a(1 − cos θ) and r = a(1 + cos θ) 8 • r = a(1 − cos θ) and r = a(1 + cos θ) ⇒ r = a and cos θ = 0 ⇒ r = a and θ = π2 + nπ ⇒ [a, π2 + nπ] ∈ intersection ⇒ n even, [a, π2 + nπ] = [a, π2 ]; n odd, [a, π2 + nπ] = [a, 3π 2 ] • Two intersection points: [a, π2 ] = (0, a) and [a, 3π 2 ] = (0, −a). • The intersection third point: the origin; but the two cardioids pass through the origin at different times (θ). Outline Contents 1 Polar Coordinates 1.1 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Relation to Rectangular Coordinates . . . . . . . . . . . . . . . . 1.3 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 4 2 Graphing in Polar 2.1 Spiral . . . . . 2.2 Limaçons . . . 2.3 Flowers . . . . 2.4 Intersections . . 6 6 6 7 8 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .