Lecture 12 - Section 9.3 Polar Coordinates Section 9.4 ...jiwenhe/Math1432/lectures/lecture12.pdfSection 9.3 Polar Coordinates Section 9.4 Graphing in Polar Coordinates Jiwen He Department

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 =y

x

r =√

x2 + y2, θ = tan−1 y

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 =y

x

r =√

x2 + y2, θ = tan−1 y

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 =y

x

r =√

x2 + y2, θ = tan−1 y

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 =y

x

r =√

x2 + y2, θ = tan−1 y

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 ⇒ y

x= 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 ⇒ y

x= 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 ⇒ y

x= 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 ⇒ y

x= 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 Limacons 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 Limacons Flowers Intersections

Limacon (Snail): r = 1− 2 cos θ

r = 0 at θ = 13π, 5

3π; |r | is a local maximum at θ = 0, π, 2π.

Sketch in 4 stages: [0, 13π], [13π, π], [π, 5

3π], [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 Limacons Flowers Intersections

Limacon (Snail): r = 1− 2 cos θ

r = 0 at θ = 13π, 5

3π; |r | is a local maximum at θ = 0, π, 2π.

Sketch in 4 stages: [0, 13π], [13π, π], [π, 5

3π], [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 Limacons Flowers Intersections

Limacon (Snail): r = 1− 2 cos θ

r = 0 at θ = 13π, 5

3π; |r | is a local maximum at θ = 0, π, 2π.

Sketch in 4 stages: [0, 13π], [13π, π], [π, 5

3π], [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 Limacons Flowers Intersections

Limacon (Snail): r = 1− 2 cos θ

r = 0 at θ = 13π, 5

3π; |r | is a local maximum at θ = 0, π, 2π.

Sketch in 4 stages: [0, 13π], [13π, π], [π, 5

3π], [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 Limacons Flowers Intersections

Limacons (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 Limacons 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 − 1

2π) = 1 + sin θ.

Rotation by 12π: r = 1 + sin(θ − 1

2π) = 1− cos θ.

Rotation by 12π: r = 1− cos(θ − 1

2π) = 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 Limacons 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 − 1

2π) = 1 + sin θ.

Rotation by 12π: r = 1 + sin(θ − 1

2π) = 1− cos θ.

Rotation by 12π: r = 1− cos(θ − 1

2π) = 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 Limacons 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 − 1

2π) = 1 + sin θ.

Rotation by 12π: r = 1 + sin(θ − 1

2π) = 1− cos θ.

Rotation by 12π: r = 1− cos(θ − 1

2π) = 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 Limacons 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 − 1

2π) = 1 + sin θ.

Rotation by 12π: r = 1 + sin(θ − 1

2π) = 1− cos θ.

Rotation by 12π: r = 1− cos(θ − 1

2π) = 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 Limacons 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 Limacons Flowers Intersections

Petal Curve: r = cos 2θ

r = 0 at θ = π4 , 3π

4 , 5π4 , 7π

4 ;|r | is a local maximum at θ = 0, π

2 , 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 Limacons Flowers Intersections

Petal Curve: r = cos 2θ

r = 0 at θ = π4 , 3π

4 , 5π4 , 7π

4 ;|r | is a local maximum at θ = 0, π

2 , 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 Limacons Flowers Intersections

Petal Curve: r = cos 2θ

r = 0 at θ = π4 , 3π

4 , 5π4 , 7π

4 ;|r | is a local maximum at θ = 0, π

2 , 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 Limacons Flowers Intersections

Petal Curve: r = cos 2θ

r = 0 at θ = π4 , 3π

4 , 5π4 , 7π

4 ;|r | is a local maximum at θ = 0, π

2 , 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 Limacons 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 Limacons 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 Limacons 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 Limacons 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 = π

2 + nπ ⇒ [a, π2 + nπ] ∈ intersection

⇒ n even, [a, π2 + nπ] = [a, π

2 ]; 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 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 Limacons 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 = π

2 + nπ ⇒ [a, π2 + nπ] ∈ intersection

⇒ n even, [a, π2 + nπ] = [a, π

2 ]; 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 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 Limacons 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 = π

2 + nπ ⇒ [a, π2 + nπ] ∈ intersection

⇒ n even, [a, π2 + nπ] = [a, π

2 ]; 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 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 Limacons 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 = π

2 + nπ ⇒ [a, π2 + nπ] ∈ intersection

⇒ n even, [a, π2 + nπ] = [a, π

2 ]; 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 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 Limacons Flowers Intersections

Outline

Polar CoordinatesPolar CoordinatesRelation to Rectangular CoordinatesSymmetry

Graphing in Polar CoordinatesSpiralLimaconsFlowersIntersections

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