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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
[email protected]://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