Polar Coordinates Graphing in Polar Coordinates

Lecture 12Section 9.3 Polar Coordinates

Section 9.4 Graphing in Polar Coordinates

Jiwen He

Department of Mathematics, University of Houston

jiwenhe@math.uh.eduhttp://math.uh.edu/∼jiwenhe/Math1432

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 1 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Polar Coordinate System

The purpose of the polarcoordinates is to representcurves that have symmetryabout a point or spiralabout a point.

Frame of Reference

In the polar coordinate system, the frame of reference is a point Othat we call the pole and a ray that emanates from it that we callthe polar axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 2 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Polar Coordinate System

The purpose of the polarcoordinates is to representcurves that have symmetryabout a point or spiralabout a point.

Frame of Reference

In the polar coordinate system, the frame of reference is a point Othat we call the pole and a ray that emanates from it that we callthe polar axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 2 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Polar Coordinate System

The purpose of the polarcoordinates is to representcurves that have symmetryabout a point or spiralabout a point.

Frame of Reference

In the polar coordinate system, the frame of reference is a point Othat we call the pole and a ray that emanates from it that we callthe polar axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 2 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Polar Coordinates

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.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 3 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

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].

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 4 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

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].

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 4 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

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].

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 4 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

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].

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 4 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Relation to Rectangular Coordinates

Relation to Rectangular Coordinates

x = r cos θ, y = r sin θ. ⇒ x2 + y2 = r2, tan θ = yx

r =√

x2 + y2, θ = tan−1y

x.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 5 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Relation to Rectangular Coordinates

Relation to Rectangular Coordinates

x = r cos θ, y = r sin θ. ⇒ x2 + y2 = r2, tan θ = yx

r =√

x2 + y2, θ = tan−1y

x.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 5 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Relation to Rectangular Coordinates

Relation to Rectangular Coordinates

x = r cos θ, y = r sin θ. ⇒ x2 + y2 = r2, tan θ = yx

r =√

x2 + y2, θ = tan−1y

x.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 5 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Relation to Rectangular Coordinates

Relation to Rectangular Coordinates

x = r cos θ, y = r sin θ. ⇒ x2 + y2 = r2, tan θ = yx

r =√

x2 + y2, θ = tan−1y

x.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 5 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

x2 + y2 = a2 ⇒ r2 = a2

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

x2 + y2 = a2 ⇒ r2 = a2

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

x2 + y2 = a2 ⇒ r2 = a2

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

x2 + y2 = a2 ⇒ r2 = a2

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

x2 + (y − a)2 = a2 ⇒ x2 + y2 = 2ay ⇒ r2 = 2ar sin θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

x2 + (y − a)2 = a2 ⇒ x2 + y2 = 2ay ⇒ r2 = 2ar sin θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

x2 + (y − a)2 = a2 ⇒ x2 + y2 = 2ay ⇒ r2 = 2ar sin θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

(x − a)2 + y2 = a2 ⇒ x2 + y2 = 2ax ⇒ r2 = 2ar cos θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

(x − a)2 + y2 = a2 ⇒ x2 + y2 = 2ax ⇒ r2 = 2ar cos θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Circles in Polar Coordinates

Circles in Polar Coordinates

In rectangular coordinates In polar coordinates

x2 + y2 = a2 r = a

x2 + (y − a)2 = a2 r = 2a sin θ(x − a)2 + y2 = a2 r = 2a cos θ

(x − a)2 + y2 = a2 ⇒ x2 + y2 = 2ax ⇒ r2 = 2ar cos θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 6 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

y = mx ⇒ yx

= m ⇒ tan θ = m

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

y = mx ⇒ yx

= m ⇒ tan θ = m

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

y = mx ⇒ yx

= m ⇒ tan θ = m

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

y = mx ⇒ yx

= m ⇒ tan θ = m

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

x = a ⇒ r cos θ = a ⇒ r = a sec θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

x = a ⇒ r cos θ = a ⇒ r = a sec θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

x = a ⇒ r cos θ = a ⇒ r = a sec θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

y = a ⇒ r sin θ = a ⇒ r = a csc θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

y = a ⇒ r sin θ = a ⇒ r = a csc θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lines in Polar Coordinates

Lines in Polar Coordinates

In rectangular coordinates In polar coordinates

y = mx θ = α with α = tan−1 m

x = a r = a sec θ

y = a r = a csc θ

y = a ⇒ r sin θ = a ⇒ r = a csc θ

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 7 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(−θ)] = cos(−2θ) = cos 2θ⇒ if [r , θ] ∈ graph, then [r ,−θ] ∈ graph⇒ symmetric about the x-axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(−θ)] = cos(−2θ) = cos 2θ⇒ if [r , θ] ∈ graph, then [r ,−θ] ∈ graph⇒ symmetric about the x-axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(−θ)] = cos(−2θ) = cos 2θ⇒ if [r , θ] ∈ graph, then [r ,−θ] ∈ graph⇒ symmetric about the x-axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(−θ)] = cos(−2θ) = cos 2θ⇒ if [r , θ] ∈ graph, then [r ,−θ] ∈ graph⇒ symmetric about the x-axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(π − θ)] = cos(2π − 2θ) = cos 2θ⇒ if [r , θ] ∈ graph, then[r , π − θ] ∈ graph⇒ symmetric about the y -axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(π − θ)] = cos(2π − 2θ) = cos 2θ⇒ if [r , θ] ∈ graph, then[r , π − θ] ∈ graph⇒ symmetric about the y -axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(π − θ)] = cos(2π − 2θ) = cos 2θ⇒ if [r , θ] ∈ graph, then[r , π − θ] ∈ graph⇒ symmetric about the y -axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(π + θ)] = cos(2π + 2θ) = cos 2θ

⇒ if [r , θ] ∈ graph, then[r , π + θ] ∈ graph⇒ symmetric about the origin.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(π + θ)] = cos(2π + 2θ) = cos 2θ

⇒ if [r , θ] ∈ graph, then[r , π + θ] ∈ graph⇒ symmetric about the origin.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Symmetry

Lemniscate (ribbon) r2 = cos 2θ

cos[2(π + θ)] = cos(2π + 2θ) = cos 2θ

⇒ if [r , θ] ∈ graph, then[r , π + θ] ∈ graph⇒ symmetric about the origin.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 8 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lemniscates (Ribbons) r 2 = a sin 2θ, r 2 = a cos 2θ

Lemniscate r2 = a sin 2θ

sin[2(π + θ)] = sin(2π + 2θ) = sin 2θ

⇒ if [r , θ] ∈ graph, then [r , π + θ] ∈ graph

⇒ symmetric about the origin.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 9 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lemniscates (Ribbons) r 2 = a sin 2θ, r 2 = a cos 2θ

Lemniscate r2 = a sin 2θ

sin[2(π + θ)] = sin(2π + 2θ) = sin 2θ

⇒ if [r , θ] ∈ graph, then [r , π + θ] ∈ graph

⇒ symmetric about the origin.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 9 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lemniscates (Ribbons) r 2 = a sin 2θ, r 2 = a cos 2θ

Lemniscate r2 = a sin 2θ

sin[2(π + θ)] = sin(2π + 2θ) = sin 2θ

⇒ if [r , θ] ∈ graph, then [r , π + θ] ∈ graph

⇒ symmetric about the origin.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 9 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Lemniscates (Ribbons) r 2 = a sin 2θ, r 2 = a cos 2θ

Lemniscate r2 = a sin 2θ

sin[2(π + θ)] = sin(2π + 2θ) = sin 2θ

⇒ if [r , θ] ∈ graph, then [r , π + θ] ∈ graph

⇒ symmetric about the origin.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 9 / 19

Polar Coordinates Graphing in Polar Coordinates Polar Coordinates Relation to Rectangular Coordinates Symmetry

Quiz

Quiz

1. r = sec θ is sym. about (a) x-axis, (b) y -axis, (c) origin.

2. r = 2 sin θ is a (a) line, (b) circle, (c) lemniscate.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 10 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Spiral of Archimedes r = θ, θ ≥ 0

The curve is a nonending spiral. Here it is shown in detail fromθ = 0 to θ = 2π.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 11 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Limaçon (Snail): r = 1− 2 cos θ

r = 0 at θ = 13π,53π; |r | is a local maximum at θ = 0, π, 2π.

Sketch in 4 stages: [0, 13π], [13π, π], [π,

53π], [

53π, 2π].

cos(−θ) = cos θ ⇒ if [r , θ] ∈ graph, then [r ,−θ] ∈ graph ⇒symmetric about the x-axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 12 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Limaçon (Snail): r = 1− 2 cos θ

r = 0 at θ = 13π,53π; |r | is a local maximum at θ = 0, π, 2π.

Sketch in 4 stages: [0, 13π], [13π, π], [π,

53π], [

53π, 2π].

cos(−θ) = cos θ ⇒ if [r , θ] ∈ graph, then [r ,−θ] ∈ graph ⇒symmetric about the x-axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 12 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Limaçon (Snail): r = 1− 2 cos θ

r = 0 at θ = 13π,53π; |r | is a local maximum at θ = 0, π, 2π.

Sketch in 4 stages: [0, 13π], [13π, π], [π,

53π], [

53π, 2π].

cos(−θ) = cos θ ⇒ if [r , θ] ∈ graph, then [r ,−θ] ∈ graph ⇒symmetric about the x-axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 12 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Limaçon (Snail): r = 1− 2 cos θ

r = 0 at θ = 13π,53π; |r | is a local maximum at θ = 0, π, 2π.

Sketch in 4 stages: [0, 13π], [13π, π], [π,

53π], [

53π, 2π].

cos(−θ) = cos θ ⇒ if [r , θ] ∈ graph, then [r ,−θ] ∈ graph ⇒symmetric about the x-axis.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 12 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Limaçons (Snails): r = a + b cos θ

The general shape of the curve depends on the relative magnitudesof |a| and |b|.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 13 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Cardioids (Heart-Shaped): r = 1± cos θ, r = 1± sin θ

Each changecos θ → sin θ → − cos θ → − sin θ

represents a counterclockwise rotation by 12π radians.

Rotation by 12π: r = 1 + cos(θ −12π) = 1 + sin θ.

Rotation by 12π: r = 1 + sin(θ −12π) = 1− cos θ.

Rotation by 12π: r = 1− cos(θ −12π) = 1− sin θ.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 14 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Cardioids (Heart-Shaped): r = 1± cos θ, r = 1± sin θ

Each changecos θ → sin θ → − cos θ → − sin θ

represents a counterclockwise rotation by 12π radians.

Rotation by 12π: r = 1 + cos(θ −12π) = 1 + sin θ.

Rotation by 12π: r = 1 + sin(θ −12π) = 1− cos θ.

Rotation by 12π: r = 1− cos(θ −12π) = 1− sin θ.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 14 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Cardioids (Heart-Shaped): r = 1± cos θ, r = 1± sin θ

Each changecos θ → sin θ → − cos θ → − sin θ

represents a counterclockwise rotation by 12π radians.

Rotation by 12π: r = 1 + cos(θ −12π) = 1 + sin θ.

Rotation by 12π: r = 1 + sin(θ −12π) = 1− cos θ.

Rotation by 12π: r = 1− cos(θ −12π) = 1− sin θ.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 14 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Cardioids (Heart-Shaped): r = 1± cos θ, r = 1± sin θ

Each changecos θ → sin θ → − cos θ → − sin θ

represents a counterclockwise rotation by 12π radians.

Rotation by 12π: r = 1 + cos(θ −12π) = 1 + sin θ.

Rotation by 12π: r = 1 + sin(θ −12π) = 1− cos θ.

Rotation by 12π: r = 1− cos(θ −12π) = 1− sin θ.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 14 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Quiz

Quiz

3. r = sin θ is the rotation byπ

2of:

(a) r = cos θ, (b) r = − sin θ, (c) r = − cos θ.

4. Today is (a) Feb. 19, (b) Feb. 20, (c) Feb. 21.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 15 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Petal Curve: r = cos 2θ

r = 0 at θ = π4 ,3π4 ,

5π4 ,

7π4 ;

|r | is a local maximum at θ = 0, π2 , π,3π2 , 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.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 16 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Petal Curve: r = cos 2θ

r = 0 at θ = π4 ,3π4 ,

5π4 ,

7π4 ;

|r | is a local maximum at θ = 0, π2 , π,3π2 , 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.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 16 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Petal Curve: r = cos 2θ

r = 0 at θ = π4 ,3π4 ,

5π4 ,

7π4 ;

|r | is a local maximum at θ = 0, π2 , π,3π2 , 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.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 16 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Petal Curve: r = cos 2θ

r = 0 at θ = π4 ,3π4 ,

5π4 ,

7π4 ;

|r | is a local maximum at θ = 0, π2 , π,3π2 , 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.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 16 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

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.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 17 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

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.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 17 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

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.

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 17 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Intersections: r = a(1− cos θ) and r = a(1 + cos θ)

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,

3π2 ]

Two intersection points: [a, π2 ] = (0, a) and [a,3π2 ] = (0,−a).

The intersection third point: the origin; but the two cardioidspass through the origin at different times (θ).

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 18 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Intersections: r = a(1− cos θ) and r = a(1 + cos θ)

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,

3π2 ]

Two intersection points: [a, π2 ] = (0, a) and [a,3π2 ] = (0,−a).

The intersection third point: the origin; but the two cardioidspass through the origin at different times (θ).

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 18 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Intersections: r = a(1− cos θ) and r = a(1 + cos θ)

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,

3π2 ]

Two intersection points: [a, π2 ] = (0, a) and [a,3π2 ] = (0,−a).

The intersection third point: the origin; but the two cardioidspass through the origin at different times (θ).

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 18 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Intersections: r = a(1− cos θ) and r = a(1 + cos θ)

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,

3π2 ]

Two intersection points: [a, π2 ] = (0, a) and [a,3π2 ] = (0,−a).

The intersection third point: the origin; but the two cardioidspass through the origin at different times (θ).

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 18 / 19

Polar Coordinates Graphing in Polar Coordinates Spiral Limaçons Flowers Intersections

Outline

Polar CoordinatesPolar CoordinatesRelation to Rectangular CoordinatesSymmetry

Graphing in Polar CoordinatesSpiralLimaçonsFlowersIntersections

Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 12 February 21, 2008 19 / 19

Polar CoordinatesPolar CoordinatesRelation to Rectangular CoordinatesSymmetry

Graphing in Polar CoordinatesSpiralLimaçonsFlowersIntersections