Introduction
Example 1
Angle increments
Example 2
Test
SKETCHING POLAR GRAPHS
GEOMETRY 3
INU0114/514 (MATHS 1)
Dr Adrian Jannetta MIMA CMath FRAS
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
Introduction
We’ve already seen how to plot points in polar coordinates to
graph a polar equation.
Do you recall how many points we needed to plot the equation
r = 3 sin 2θ ? (A lot!)
In this presentation we’ll see an alternative method for sketching
polar curves. To graph the function
r = f (θ )
first we’ll sketch the graph y = f (θ ) on Cartesian axes. This is
usually much easier.
Then we’ll use the Cartesian graph to construct the polar graph.
And we’ll do all of this without polar graph paper to help!
Let’s go straight to some examples.
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
Sketching a polar curve
Sketch the curve r = 1 + cos θ , 0 ≤ θ ≤ 2π.
Begin by making a Cartesian sketch of r against θ .
The graph is
r simply the graph of cos θ but shifted up one unit.
2
1
θ
π
2
3π
2
π
2π
We’ll refer to this graph as a guide and build the polar graph in
stages.
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Look at the curve between the
points denoted by ① and ②.
Example 2
Test
r
2
①
②
1
As the angle θ increases from 0
to π2 , then r decreases from 2 to
1.
θ
π
2
3π
2
π
2π
We’ll transfer this information polar axes.
Start at the point (2; 0) and end at (1; π2 ).
Draw a curve which turns through the
angle from 0 to π2 .
While we draw the curve we take care to
decrease the distance from the origin
from 2 to 1.
θ=
1
π
2
②
① θ =0
θ =π
1
2
−1
Notice that r > 1 for all points on this
part of the curve.
Sketching polar graphs
θ=
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3π
2
Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
r
Now, look at the curve between
the points ② and ③.
2
①
②
1
As angle θ increases from π2 to
π, then r decreases from 1 to 0.
③
π
2
θ
3π
2
π
2π
Again we transfer this behaviour to the polar axes.
Continuing from (1; π2 ) and end at (0; π).
Draw a curve which turns through the
angle from π2 to π.
The curve is dragged towards the origin
as the angle approaches π.
θ=
1
θ =π
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②
① θ =0
③
1
2
−1
θ=
Sketching polar graphs
π
2
3π
2
Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
r
Examine the guide graph
between ③ and ④.
2
①
②
1
As θ increases from π to
r increases from 0 to 1.
3π
2
④
then
③
π
2
θ
3π
2
π
2π
Go to the polar axes and sketch the next section of the curve.
Continue from (0; π) and end at (1; 3π
2 ).
Draw a curve which turns through the
angle from π to 3π
2 .
The curve is pulled outwards from the
origin as the angle approaches 3π
2 .
θ=
1
θ =π
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②
① θ =0
③
1
−1
θ=
Sketching polar graphs
π
2
2
④
3π
2
Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
r
The last section of the graph
between ④ and ⑤.
2
⑤
①
②
1
As θ increases from 3π
2 to 2π
then r increases from 1 to 2.
④
③
π
2
θ
3π
2
π
2π
Go to the polar axes and sketch the final part of the curve.
Continue from (1; 3π
2 ) and end at (2; 2π).
Draw a curve which turns through the
angle from 3π
2 to 2π.
The curve is pulled outwards from 1 to
2 as the angle increases to 2π.
θ=
1
θ =π
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②
③
1
−1
θ=
Sketching polar graphs
π
2
2
①
⑤ θ =0
④
3π
2
Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
Here is the final version of the graph r = 1 + cos θ , 0 ≤ θ ≤ 2π.
1
1
2
−1
This is a cardioid (heart-shaped) curve.
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
Some advice: angle increments
In the first example we plotted a series of points at intervals of π2
and then inferred the behaviour of the curve between those points.
Generally, when plotting polar graphs we want to know which
angles are valuable for decoding the shape of the of the curve.
A useful strategy for an equation with a multiple angle, e.g.,
r = 3 sin 2θ
r = 4 cos 3θ
is to set nθ = π2 .
For the two examples above: we would look for angle increments
of π4 for the first, and π6 for the second.
We’ll see how this works on the next example.
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
Polar graph sketch
Sketch the graph of the function r = 4 sin 3θ .
The Cartesian graph of the function can obtained from the sine curve:
stretch vertically by a factor of 4 and the period decreased by a factor of 3.
You might want to take this opportunity to revise trig transformations!
Here is the Cartesian guide graph:
r
4
π
6
π
3
π
2
2π
3
5π
6
π
θ
−4
The useful angles to consider on the polar graph are those where 3θ = π2 ,
in other words, increments of π6 .
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
r
We’re going to build up the polar sketch by
examining each section of the Cartesian
graph.
4
π
6
Here is the Cartesian graph again for easy
reference!
π
3
π
2
5π
6
2π
3
θ
π
−4
π
2
Let’s draw the polar axes.
Mark lines at angle increments of π6 .
Guide graph: between 0 and
increases from 0 to 4.
π
3
2π
3
π
6:
π
6
5π
6
r
π
0
Polar graph: start at (0; 0) and end at
(4; π6 ).
Draw the first section of the curve as
shown.
7π
6
11π
6
4π
3
5π
3
3π
2
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
r
We’re going to build up the polar sketch by
examining each section of the Cartesian
graph.
4
π
6
Here is the Cartesian graph again for easy
reference!
π
3
π
2
5π
6
2π
3
θ
π
−4
π
2
Let’s take a look at the next section.
Guide graph: between
decreases from 4 to 0.
π
6
and π3 : r
Polar graph: start at (4; π6 ) and end at
(0; 0).
The symmetry of the sine curve
suggests this is going to be a
reflection of the first curve we drew!
π
3
2π
3
π
6
5π
6
π
0
7π
6
11π
6
5π
3
4π
3
3π
2
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
r
We’re going to build up the polar sketch by
examining each section of the Cartesian
graph.
4
π
6
Here is the Cartesian graph again for easy
reference!
π
3
π
2
2π
3
5π
6
θ
π
π
2
π
3
2π
3
π
6
5π
6
to π2 : r goes 0 → −4.
Polar graph: start (0; π3 ) and end
(−4; π2 ).
π
3
−4
On the next part of the guide graph,
r becomes negative — we plot
directly away from that angle. Add
π!
Guide graph:
Test
π
0
So: curve goes from origin to (4; 3π
2 ).
Then the guide graph goes to zero
between π2 and 2π
3 ; the polar graph
decreases from 4 to zero between 3π
2
and 5π
3 .
Sketching polar graphs
7π
6
11π
6
4π
3
5π
3
3π
2
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
r
We’re going to build up the polar sketch by
examining each section of the Cartesian
graph.
4
π
6
Here is the Cartesian graph again for easy
reference!
2π
3
to
5π
6 :
π
2
5π
6
2π
3
θ
π
π
2
π
3
2π
3
r goes 0 → 4.
Polar graph: start (0; 2π
3 ) and end
(4; 5π
6 ).
π
3
−4
On this section of the guide graph, r
is positive again.
Guide graph:
Test
π
6
5π
6
π
0
The guide graph shows r going
4 → 0 in the final interval.
The polar graph goes 4 → 0 between
5π
6 and π.
7π
6
11π
6
5π
3
4π
3
3π
2
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
If we try to continue following the guide graph for angles larger than π
we find the polar curve begins to repeat and we simply retrace the
sections already drawn.
Here is the final curve for r = 4 sin 3θ , 0 ≤ θ ≤ π.
π
2
π
3
2π
3
π
6
5π
6
π
0
7π
6
11π
6
4π
3
5π
3
3π
2
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
Test yourself
If you’ve read and understood the examples in these notes, you
should be able to draw the following polar curves.
Consider the functions
1
r = 3 sin 2θ
2
r = 1 + 2 cos θ
In each case
• Sketch the Cartesian guide graph.
• Use the guide graph to plot the polar graph.
The answers are shown on the next two slides.
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
Guide graph for r = 3 sin 2θ .
r
3
π
4
π
2
3π
4
5π
4
π
3π
2
7π
4
2π
θ
−3
We need to put angle increments of 2θ =
π
2
(so θ = π4 ) on the polar graph.
π
2
π
4
3π
4
π
0
5π
4
7π
4
3π
2
Sketching polar graphs
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Adrian Jannetta
Introduction
Example 1
Angle increments
Example 2
Test
Guide graph for r = 1 + 2 cos θ . Dashed lines to show r values more clearly.
r
3
1
π
2
3π
2
π
θ
2π
−3
In this case angle increments of θ =
π
2
will be fine on the polar graph.
π
2
1
1
π
3
0
−1
3π
2
Sketching polar graphs
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Adrian Jannetta