7 Polar Coordinates

8/18/2019 7 Polar Coordinates

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POLAR COORDINATES AND

POLAR GRAPHS



At the end of the lecture, you should be able to: Define the coordinates of points using polar

coordinate system.

Translate the rectangular coordinates to polarcoordinates and vice versa.

Draw graphs of equations in polar coordinateaxis.



THE POLAR COORDINATE SYSTEM

• There are various types of coordinate system.

• The rectangular system is probably the most

important. In this system a point is located by its

distances from two perpendicular lines.• The second type of coordinate system is called

polar coordinates system.

• It is a coordinate system in which the coordinates

of a point in a plane are its distances from a fixed

point and its direction from a fixed line.




The reference frame in the polar coordinate

system is a half-line drawn from some point in theplane.

The half-line is represented by OA.

The point O is called the origin or pole and OA isthe polar axis.




The position of any point P in the plane is determinedby the distance OP and the angle AOP.

The segment OP, denoted by r , is referred to as radius

vector; The angle AOP, denoted by θ , is called the vectorial

angle.

The coordinates of P is written as P(r, θ ) or just (r, θ ).



The center of the graph

is called the pole.

Angles are measured

from the positive x axis.

Points arerepresented by a

radius and an angle

To plot the point

First find the angle

Then move out along

the terminal side 5units

polar axis

terminal side

(r,

)



A negative angle would be measured clockwise

like usual.

To plot a point with

a negative radius,

find the terminal

side of the anglebut then measure

from the pole in

the negative

direction of theterminal side.



Let's plot the following points:

We can see thatunlike in the

rectangular

coordinate system,

there are manyways to list the

same point.



RELATIONS BETWEEN RECTANGULAR AND POLAR

COORDINATES




COORDINATES

• To obtain r and θ in terms of x and y, we use thePythagorean theorem and the tangent function. That is

• But the range of tan-1 is –π/2 < θ < π/2 so the value of

θ from the previous equation will not represent any

point to the left of the y-axis. So the measure of the

angle θ is given by

x

yand y xr tan

222

0tan

0tan

1

1

xif

x

y

xif x

y




COORDINATES




COORDINATES

Rectangular Form:

Polar Form:

C By Ax

sincos B A

C r




COORDINATES



POLAR EQUATIONS and POLAR GRAPHS



Graph of r = a

In general, the graph of r = a is

a circle with center at the pole.



Graph of = a

In general, the graph of = a

is a line through the origin.




• Graph

a. r= 7cosθ

b. r= -4sinθ



POLAR GRAPHS



• The heart-shaped graph is a cardioid.

• In general, the graph of any of the polar equationsbelow, with a ≠ 0, is a cardioid.

POLAR GRAPHS



• The heart-shaped graph is a cardioid.

• In general, the graph of any of the polar equationsbelow, with a ≠ 0, is a cardioid.

POLAR GRAPHS

Th h i di id



Let each unit be 1/4.

The graph is a cardioid .

θ r

0 1

π/2 0π 1

3π/2 2




•

Graphr = 2 + 2cosθ

θr

0 4

π/2 2

π 03π/2 2


The graph is a cardioid .




•

Graphr = 2 + 2cosθ

θ r

0 4

π/6 3.73

π/3 3

π/2 2

2π/3 1

5π/6 0.27

π 0


θ r

7π/6 0.27

4π/3 1

3π/2 2

5π/3 3

11π/6 3.73

2π 4



POLAR GRAPHS

The graph is a limacon with a dimple.



Let each unit be 1.

The graph is a limacon with a dimple.

θ r

0 5

π/2 3

π 1

3π/2 3



Graphs of Polar Equations

Limacon with an inner loop

θ r

0 6

π/2 2

π -2

3π/2 2

Let each unit be 1.




4

2

2

4

5

Limacon with an inner loop




Graph

r = 3 + sinθ

Convex Limacon


θr

0 3

π/2 4

π3

3π/2 2



POLAR GRAPHS




Four – leafed rose

2

1

1

2

2 2

A




Graph r= -2sin(3θ)

6,2

r If

3

2

#

2: petalsof

Interval

2

3,2,

6

5,2,

6,2

:

petalstheof Tip



POLAR GRAPHS



POLAR GRAPHS

Lemniscates/ Lemniscate of Bernoulli

r²=-a²sin2θ r²=-a²cos2θ

values of θ that make the right member positive

are excluded:

for r²=-a²sin2θ, the excluded values are

0<θ

<π

/2 andπ

<θ

<3π

/2for r²=-a²cos2θ, the excluded values are

-π/4< θ < π/4 and 3π/4< θ < 5π/4

The graph is a lemniscate




The excluded values

are π/2< θ < π and3π/2< θ < 2π




lemniscate

2

2

G h f P l E ti




• Graph

r² = -9 cos2θ

the excluded values

are -π/4< θ < π/4

and 3π/4< θ < 5π/4

The graph is a lemniscate

7 Polar Coordinates

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