Polar coordinates
Anushaya Mohapatra
Department of Mathematics
BITS PILANI K K Birla Goa Campus, Goa
November 21, 2022
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Polar coordinates
To define polar coordinates for a point . Fix an origin O, called the pole,
and an initial ray from O. (initial ray is called polar axis).(r , θ) represent
the polar coordinate of the point P, where r is the directed distance from
O to P andθ is the directed angle from the initial ray to OP.
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Conventions
Positive θ: It is measured in the counterclockwise
direction from the polar axis.
Negative θ: It is measured in the clockwise direction
from the polar axis.
Positive r: It is measured in the same direction of the
initial ray OP form the origin O.
Negative r: It is measured in the reverse direction of
the initial ray OP from the origin O.
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Polar coordinates
If r > 0, the point (r , θ) lies in the same quadrant as θ.
If r < 0, the point (r , θ) lies on the opposite side of the pole.
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Examples
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Examples
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Examples
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Cartisian vs Polar
In cartisian system, each point has only one
representation, however in polar system each point has
many representation.
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Cartisian and Polar
If the point P has Cartesian coordinates (x, y ) and polar
coordinates (r , θ), then we have
x = r cos(θ)
y = r sin(θ)
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Polar to Cartesian
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Example
Convert the point (2, π/3) from polar to Cartesian
coordinates:
So (1,
√
3) is the cartesian coordinate.
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Examples
1
Represent the point with Cartesian coordinates
(−1, −1) in terms of polar coordinates.
p
√
r = x2 + y2 = 2
y
π
= 1, θ =
x
4
√
Thus, one possible answer is: ( 2, π/4) Is the answer
correct ?
tanθ =
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Polar Curves
The graph of a polar equation r = f (θ) consists of all
points that have at least one polar representation (r , θ),
whose coordinates satisfy the equation.
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Examples
1. What curve is represented by the polar equation r = 4?
2. Sketch the polar curve θ = 1.
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Polar curve
The curve r = 4 represents the circle with center O and
radius 4.
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polar curve
The curve θ = 1 is the straight line that passes through O
and makes an angle of 1 radian with the polar axis.
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Polar curve sketching
Example 1: Sketch the curve with polar equation
r = 2 cos(θ). Find a Cartesian equation for this curve.
Ans: It is a circle.
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Example 1
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Example 1b
To convert the given equation to a Cartesian equation, we use the
following:
x = r cos(θ) = 2x/r
which gives: 2x = r 2 = x 2 + y 2 or x 2 + y 2 − 2x = 0 Or the equation is
(x − 1)2 + y 2 = 1
which represent a circle with center (1, 0) and radius 1.
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Example 2
Sketch the curve r = 1 + sin(θ). We first sketch the
graph in cartesian coordinates.
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We see that, as θ increases from 0 to π/2, r (the distance from O)
increases from 1 to 2. So, we sketch the corresponding part of the polar
curve as follows:
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Putting together the various parts of the curve, we sketch the complete
curve as shown next.
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Putting together the various parts of the curve, we sketch the complete
curve as shown next. It’s called the cardioid.
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Example 3
Sketch the curve r = cos(2θ).
We first sketch in cartesian coordinates.
the figure shows that as θ increases from 0 to π/4, r
decreases from 1 to 0. It shows how r varies for
0 ≤ θ ≤ 2π.
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Example-3
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Symmetry
Symmetry about x-axis: If the point (r , θ) lies on the
graph, then the point (r , θ) or the point (−r , π − θ) also
lies on the graph.
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Symmetry about x-axis
The polar curve r = 1 + cos(θ) is symmetric about x−axis.
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Symmetry about y-axis
If the point (r , θ) lies on the graph, then the point
(r , θ) or the point (−r , π − θ) also lies on the graph.
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Symmetry about y-axis
The polar curve r = 1 + sin(θ) is symmetric about y −axis.
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Symmetry about origin
If the point (r , θ) lies on the graph, then the point
(−r , θ) or the point (r , π + θ) also lies on the graph.
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Symmetry about origin
r = cos(2θ)
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Polar Curves: Cardioids
a
a : 1 + cos θ, b : 1 + sin θ
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b
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Limacons
a
b
a : c + d sin θ(c > d) b : c + d cos θ(c < d)
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Roses
a
b
a : 4 cos(5θ) b : 4 cos(4θ)
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Roses
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Spirals
r = ±θ, r = e θ
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Tangent to polar curves
To find a tangent line to a polar curve r = f (θ), we regard
θ as a parameter and write its parametric equations as:
x = r cos θ = f (θ) cos θ
y = r sin θ = f (θ) sin θ
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Tangent to polar curve
To find the slope of the tangent to parametric curves, we
have product rule:
dy
dy /dθ
dr /dθ sin θ + r cos θ
=
=
dx
dx/dθ
dr /dθ cos θ − r sin θ
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Tangent to the polar curves
Horizontal tangets can be found by setting dy /dθ = 0
provided dx/dθ ̸= 0
Vertical tangets can be found by setting dx/dθ = 0
provided dy /dθ ̸= 0
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Example 4
1
2
Find the slope of the tangent line for the cardioid
r = 1 + sin(θ) at θ = π/3
Find the points on the cardioid where the tangent line
is horizontal or vertical.
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Thank you
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