Polar Coordinate System CALCULUS-III

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Title: Polar Coordinate System CALCULUS-III

1
Polar Coordinate SystemCALCULUS-III

Dr. Farhana Shaheen

2
Polar Coordinate System

In mathematics, the polar coordinate system is a
two-dimensional coordinate system in which each
point on a plane is determined by a distance from
a fixed point and an angle from a fixed
direction.
The fixed point (analogous to the origin of a
Cartesian system) is called the pole, and the ray
from the pole with the fixed direction is the
polar axis. The distance from the pole is called
the radial coordinate or radius, and the angle is
the angular coordinate, polar angle, or azimuth.

3
2-D (Plane) Polar Coordinates

Thus the 2-D polar coordinate system involves the
distance from the origin and an azimuth angle.
Figure 1 shows the 2-D polar coordinate system,
where r is the distance from the origin to point
P, and ? is the azimuth angle measured from the
polar axis in the counterclockwise direction.
Thus, the position of point P is described as (r,
? ). Here r ? are the 2-D polar coordinates.

4
Figure 1

Any point P in the plane has its position in the
polar coordinate system determined by (r, ?).

5
Some Points With Their Polar Coordinates
6
Rectangular and Polar Coordinates

Rectangular coordinates and polar coordinates are
two different ways of using two numbers to locate
a point on a plane.
Rectangular coordinates are in the form (x, y),
where 'x' and 'y' are the horizontal and vertical
distances from the origin.

7
A point in Cartesian Plane
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Polar coordinates

Polar coordinates are in the form (r, ?), where
'r' is the distance from the origin to the point,
and ? is the angle measured from the positive 'x'
axis to the point

10
Relation between Polar and Rectangular
Coordinates

To convert between polar and rectangular
coordinates, we make a right triangle to the
point (x, y), like this

11
The relationship between Polar and Cartesian
coordinatesx r Cos ?, y r Sin ?
12

1. Polar to Rectangular
From the diagram above, these formulas convert
polar coordinates to rectangular coordinates
x r cos ?, y r sin ?.
So the polar point (r, ?) can be converted to
rectangular coordinates as
(x, y) ( r cos ?, r sin ?)
Example A point has polar coordinates
(5, 30º). Convert to rectangular coordinates.
Solution (x, y) (5cos30º, 5sin30º)
(4.3301, 2.5)

13
Converting between polar and Cartesian
coordinates

The two polar coordinates r and ? can be
converted to the Cartesian coordinates x and y by
using the trigonometric functions sine and
cosine
while the two Cartesian coordinates x and y can
be converted to polar coordinate r ? , using
the Pythagorean theorem) as follows

2. Rectangular to Polar
From the diagram below, these formulas convert
rectangular coordinates to polar coordinates
By the rule of Pythagoras
r2 x2 y2.
Also, Tan ? y/x implies
? tan-1( y/x )
So the rectangular point (x,y) can
be converted to polar coordinates
like this
( r,?) ( r, tan-1( y/x ) )

15
To plot a point in Polar Coordinate

We first mark the angles, in the anti-clockwise
direction from the polar axis.

16
Negative Distance
17
OQ is extension of OP

With coordinates P(r,?) and Q(-r, ?p)

For any real r gt 0 and for all integers k

19
A planimeter, which mechanically computes polar
integrals

A planimeter is a measuring instrument used to
determine the area of an arbitrary
two-dimensional shape.

20
Cartesian equations of Parabolas

Move the original graph yx2 up 2 units. The
resultant graph is y x22

21
Polar and Cartesian equations of a Parabola
22
Polar and Cartesian equations of a Parabola
23
Example

Find the polar equation of each of the following
curves with the given Cartesian equation
a) x c
B) x2 y y3 4

24
Solution
25
To convert Cartesian equation into polar equation

Example

26
Polar Equations of Straight Lines

? a, for any fixed angle a.
Exp ? p/4

27
Straight Lines

Standard equation
of straight line
in Cartesian coordinates
y mx c

28
Polar Equations of Straight Lines

r Cos ? k or r k Sec ?.
It is a vertical line through k.
It is equivalent to the Cartesian equation
x k.
r Sin ? k or r k Csc ?.
It is a horizontal line through k.
It is equivalent to the Cartesian equation
y k.

29
Polar equation of a curve

The equation defining an algebraic curve
expressed in polar coordinates is known as a
polar equation. In many cases, such an equation
can simply be specified by defining r as a
function of ?. The resulting curve then consists
of points of the form (r(?), ?) and can be
regarded as the graph of the polar function r.
Different forms of symmetry can be deduced from
the equation of a polar function r. If r(-?)
r(?) the curve will be symmetrical about the
horizontal (0/180) ray, if r(p - ?) r(?) it
will be symmetric about the vertical (90/270)
ray, and if r(? - a) r(?) it will be
rotationally symmetric a counterclockwise about
the pole.
Because of the circular nature of the polar
coordinate system, many curves can be described
by a rather simple polar equation, whereas their
Cartesian form is much more intricate. Among the
best known of these curves are the polar rose,
Archimedean spiral, lemniscates, limaçon, and
cardioid.

30
Curve shapes given by polar equations

There are many curve shapes given by polar
equations. Some of these are circles, limacons,
cardioids and rose-shaped curves.
Limacon curves are in the form
r a b sin(?) and r a b cos(?)
where a and b are constants.
Cardioid (heart-shaped) curves are special curves
in the limacon family where a b.
Rose petalled curves have polar equations in the
form of r a sin(n?) or r a cos(n?) for ngt1.
When n is an odd number, the curve has n petals
but when n is even the curve has 2n petals.

31
Polar Equations of Circles

r k A circle of radius k centered at the
origin.
r a sin ? A circle of radius a, passing
through the origin. If a gt 0, the circle will be
symmetric about the positive y-axis if a lt 0,
the circle will be symmetric about the
negative y-axis.
r a cos ? A circle of radius a, passing
through the origin. If a gt 0, the circle will be
symmetric about the positive x-axis if a lt 0,
the circle will be symmetric about the negative
x-axis.

32
Equations of Circle

A circle with equation r(?) 1
The general equation for a circle with a center
at (r0, f) and radius a is
This can be simplified in various ways, to
conform to more specific cases, such as the
equation
for a circle with a center at the pole and radius
a.

33
A circle with equation r(?) 1
34
Parametric Equation of a Circle

For a circle with origin (h,k) and radius
r x(t) r cos(t) h y(t) r
sin(t) k

35
Graph Polar Equations

Step 1
Consider r 4 sin(?) as an example to learn how
to graph polar coordinates.
Step 2
Evaluate the equation for values of (?) between
the interval of 0 and p. Let ? equal 0, p /6 , p
/4, p /3, p /2, 2p /3, 3p /4, 5p /6 and p.
Calculate values for r
by substituting these values into the equation.
Step 3
Use a graphing calculator to determine the values
for r. As an example, let
? p /6. Enter into the calculator 4 sin(p /6).
The value for r is 2 and the point
(r, ?) is (2, p /6). Find r for all the (?)
values in Step 2.
Step 4
Plot the resulting (r, ? ) points from Step 3
which are (0,0), (2, p /6), (2.8, p /4), (3.46,p
/3), (4,p /2), (3.46, 2p /3), (2.8, 3p /4), (2,
5p /6), (0, p) on graph paper and connect these
points. The graph is a circle with a radius of 2
and center at (0, 2). For better precision in
graphing, use polar graph paper.

36
Simplify the Graphing of Polar Equations

Look for symmetry when graphing these functions.
As an example use the polar equation r4 sin?.
You only need to find values for ? between p (Pi)
because after p the values repeat since the sine
function is symmetrical.
Step 2
Choose the values of ? that makes r maximum,
minimum or zero in the equation. In the example
given above r 4 sin (?), when ? equals 0 the
value for r is 0. So (r, ?) is (0,0). This is a
point of intercept.
Step 3
Find other intercept points in a similar manner.

37
Graphing Polar Equations

Example 1 Graph the polar equation given by
r 4 cos t
and identify the graph.

38
Solution

We first construct a table of values using the
special angles and their multiples. It is useful
to first find values of t that makes r maximum,
minimum or equal to zero. r is maximum and equal
to 4 for t 0. r is minimum and equal to -4 for
t p and r is equal to zero for t p/2.

39
Plotting of points in polar coordinates
40
Join the points drawing a smooth curve r 4 cos
t
41
Limacon

In geometry, a limaçon, also known as a limaçon
of Pascal, is defined as a roulette formed when a
circle rolls around the outside of a circle of
equal radius. It can also be defined as the
roulette formed when a circle rolls around a
circle with half its radius so that the smaller
circle is inside the larger circle. Thus, they
belong to the family of curves called centered
trochoids more specifically, they are
epitrochoids. The cardioid is the special case in
which the point generating the roulette lies on
the rolling circle the resulting curve has a
cusp.

42
Construction of a limacon
43
Polar Equations of Limacons

Equations of limacons have two general forms
r a b sin ? and r a b cos ?
Depending on the values of a and b, the graph
will take on one of three general shapes and will
either pass through the origin or not as
summarized below.

44
Equations of limacon

r a b Cos ? r a b Sin ?
If a gt b then you have a dimple
If a b then you have a cardioid
If a lt b then you have an interior lobe.

45
Graphs of Limacons

a gtb a b
a lt b

46
Cardioids

When a b, the graph has a rounded \heart"
shape, with the pointed (convex) indentation of
the heart located at the origin. Such a graph is
called a cardiod. They may be categorized as
follows
r a(1 sin ?) . Symmetric about the positive
y-axis if symmetric about the
negative y-axis if - '.
r a(1 cos ?) . Symmetric about the positive
x-axis if ' symmetric about the negative
x-axis if -'.
In either case, the pointed \heart" indentation
will point in the direction of the axis of
symmetry. The maximum distance of the graph from
the origin will be 2a and the point furthest
away from the origin will lie on the axis of
symmetry.

47
Limacons
48
Dimpled Limacons

r3/2cos(t) (purple)r'3/2-sin(t) (red)

49
If a lt b then you have an interior lobe in
Limacon
50
The family of limaçons is varied by making a
range from -2 to 2, and then back to -2 again.
51
Limacon Pedal curve of a circle
52
Graph for the equationr 2 2 sin t (Cardiod)

t 0, r 2
t p/6,r 3.0
t p/4,r 3.4
t p/3,r 3.7
t p/2,r 4
t 2p/3,r 3.7
t 3p/4,r 3.4
t p,r 2

53
Cardioid

r1cos(t)

Changing b to - b has the same effect on the
cardiod as with the other limacons that is a
reflection occurs.
r1cos(t) (magenta)r1- cos(t) (purple)

55
If a b, the cardioid will increase or
decrease in size depending on the value of a and
b

r0.50.5cos(t) (black)r22cos(t)
(purple)r33cos(t) (red)r44cos(t) (blue)

56
Rose curves

r a Sin n?
r a Cos n?
where n gt 1.
Graph has n petals
if n is odd, and 2n
petals if n is even.

57
Polar rose

A polar rose is a famous mathematical curve that
looks like a petalled flower, and that can be
expressed as a simple polar equation
r a Sin n?
r a Cos n?, for n gt 1.
If n is an integer, these equations will produce
an n-petalled rose if n is odd, or a 2n-petalled
rose if n is even. If n is rational but not an
integer, a rose-like shape may form but with
overlapping petals. Note that these equations
never define a rose with 2, 6, 10, 14, etc.
petals. The variable a represents the length of
the petals of the rose.

58
A polar rose with equation r(?) 2 sin 4?
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Graph for the equation r 4 cos 2t

t 0, r 4
t p/6,r 2
t p/4,r 0
t p/2,r -4
t p/3,r -2
t 2p/3,r -2
t 3p/4,r 0
t p,r 4

61
Pretty PetalsRoses

Consider the following polar equationsr cos
(2 t) (light red)r 3 cos (2 t) (heavy
red)and their associated graphs.

62
The number of leaves is determined by n.

r 5 cos (8 t)

r 2 cos (3 t) r 3 cos (5 t) r 4 cos(7 t)

r 2 cos (3 t) (blue)r 2 sin (3 t) (purple)

Examples of flowers

66
Three-petal flowers
67
Four- petal flowers

Why is a four-petalled flower considered lucky?
A FOUR LEAF CLOVER One leaf for fame,One leaf
for wealth,And one leaf for a faithful
lover,And one leaf to bring glorious health
There are many legends about this small plant.
One is that Eve took a four-leaf clover with her
when leaving
the Garden of Eden. This would make
it a very rare plant indeed, and very lucky.

68
Five-petal flowers
69
Spirals

Logarithmic Spiral The logarithmic spiral is a
spiral whose polar equation is given by r aeb?,
where r is the distance from the origin, ? is the
angle from the polar-axis, and a and b are
arbitrary constants. The logarithmic spiral is
also known as the growth spiral, equiangular
spiral, and spira mirabilis.

70
Logarithmic spiral

A logarithmic spiral, equiangular spiral or
growth spiral is a special kind of spiral curve
which often appears in nature. The logarithmic
spiral was first described by Descartes and later
extensively investigated by Jakob Bernoulli, who
called it Spira mirabilis, "the marvelous spiral".

71
Logarithmic Spiral r a b?

The distance between successive coils of a
logarithmic spiral is not constant as with the
spirals of Archimedes.

72
Spirals of Archimedes

Polar graphs of the form r a? b where a is
positive and b is nonnegative are called Spirals
of Archimedes. They have the appearance of a coil
of rope or hose with a constant distance between
successive coils.

73
Archimedean spiral

The Archimedean spiral (also known as the
arithmetic spiral) is a spiral named after the
3rd century BC Greek mathematician Archimedes. It
is the locus of points corresponding to the
locations over time of a point moving away from a
fixed point with a constant speed along a line
which rotates with constant angular velocity.
Equivalently, in polar coordinates (r, ?) it can
be described by the equation
r a? b
with real numbers a and b. Changing the parameter
a will turn the spiral, while b controls the
distance between successive turnings.

74
The Archimedean spiral
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The logarithmic spiral can be distinguished from
the Archimedean spiral by the fact that the
distances between the turnings of a logarithmic
spiral increase in geometric progression, while
in an Archimedean spiral these distances are
constant.

77
Hyperbolic spiral

Any polar equation that has the form r a/?
where agt0 is a hyperbolic spiral.

78
Cornu Spiral in Complex Plane

A plot in the complex plane of the points
B(t)S(t)iC(t)

79
Fermat's Spiral

Fermat's spiral, also known as the parabolic
spiral, is an Archimedean spiral having polar
equation r2 a2?

80
Spirals in Nature

In nature, you may have noticed that shells of
some sea creatures are shaped like logarithmic
spirals particularly the nautilus.

81
Spirals in Nature
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Spiral galaxies
84
Spiral galaxies

Galaxies, by contrast, rotate either direction
depending on your point of view -- there is no
known up or down in the universe. (A study in the
late 1990s suggested the universe was
directional, but the work was soon refuted.)
Why do galaxies rotate in the first place? The
answer goes back to the formation of the
universe, when matter raced outward in all
directions. Clumps eventually formed, and these
clumps began to interact gravitationally. Once
stuff moved off a straight course and began to
curve toward something else, angular momentum, or
spin, set in. The laws of physics say angular
momentum must be conserved.

Astronomers don't know exactly how a galaxy like
the Milky Way gets its spiral arms. But the
basics are understood. Gravitational disturbances
called density waves, rippling slowly through a
galaxy, are thought to cause it to wind up and
generate the spiral appearance.
The spiral arms of a galaxy are places where gas
piles up at the wave crests. The material does
not move with the spirals, but rather is caught
up in them.

86
The Whirlpool Galaxy

The arms of spiral galaxies often have the shape
of a logarithmic spiral, e.g.
Whirlpool
Galaxy

The spirals show the places where newly born
stars reside, while older stars reside in the
core of this spiral galaxy depicting that the
arms are star forming factories. When you look at
the upper right portion of above picture, you
notice that another galaxy called NGC 5195
appears to be tugging the arms of whirlpool
galaxy but latest images have shown that it is
passing behind this galaxy.

88
Curves that are close to being logarithmic spirals

In several natural phenomena one may find curves
that are close to being logarithmic spirals. Here
follows some examples and reasons
The approach of a hawk to its prey. Their
sharpest view is at an angle to their direction
of flight this angle is the same as the spiral's
pitch.4
The approach of an insect to a light source. They
are used to having the light source at a constant
angle to their flight path. Usually the sun (or
moon for nocturnal species) is the only light
source and flying that way will result in a
practically straight line.
The arms of spiral galaxies. Our own galaxy, the
Milky Way, is believed to have four major spiral
arms, each of which is roughly a logarithmic
spiral with pitch of about 12 degrees, an
unusually small pitch angle for a galaxy such as
the Milky Way. In general, arms in spiral
galaxies have pitch angles ranging from about 10
to 40 degrees.
The nerves of the cornea.
The arms of tropical cyclones, such as
hurricanes.

89
Spirals Down the drain

Back home, all of this has almost nothing to do
with your bathtub drain, which creates another
spiral shape.
There is a popular myth, though, owing to the
rotational direction of a hurricane, that says
the water in bathtubs rotates a certain direction
in the Northern Hemisphere.
It's not true.

90
Romanesco broccoli
91
Mandelbrot set

A section of the Mandelbrot set following a
logarithmic spiral. The Mandelbrot set, named
after Benoît Mandelbrot,
is a set of points in the
complex plane,
the boundary of which
forms a fractal.

92
Applications

Polar coordinates in two-dimensional space can be
used only where point positions lie on a single
two-dimensional plane. They are most appropriate
in any context where the phenomenon being
considered is inherently tied to direction and
length from a center point. For instance, the
examples above show how elementary polar
equations suffice to define curvessuch as the
Archimedean spiralwhose equation in the
Cartesian coordinate system would be much more
intricate. Moreover, many physical systemssuch
as those concerned with bodies moving around a
central point or with phenomena originating from
a central pointare simpler and more intuitive to
model using polar coordinates. The initial
motivation for the introduction of the polar
system was the study of circular and orbital
motion.

93
Position and navigation

Polar coordinates are used often in navigation,
as the destination or direction of travel can be
given as an angle and distance from the object
being considered. For instance, aircraft use a
slightly modified version of the polar
coordinates for navigation. In this system, the
one generally used for any sort of navigation,
the 0 ray is generally called heading 360, and
the angles continue in a clockwise direction,
rather than counterclockwise, as in the
mathematical system. Heading 360 corresponds to
magnetic north, while headings 90, 180, and 270
correspond to magnetic east, south, and west,
respectively.22 Thus, an aircraft traveling 5
nautical miles due east will be traveling 5 units
at heading 90 (read zero-niner-zero by air
traffic control).23

94
Modeling

Systems displaying radial symmetry provide
natural settings for the polar coordinate system,
with the central point acting as the pole. A
prime example of this usage is the groundwater
flow equation when applied to radially symmetric
wells. Systems with a radial force are also good
candidates for the use of the polar coordinate
system. These systems include gravitational
fields, which obey the inverse-square law, as
well as systems with point sources, such as radio
antennas.
Radially asymmetric systems may also be modeled
with polar coordinates. For example, a
microphone's pickup pattern illustrates its
proportional response to an incoming sound from a
given direction, and these patterns can be
represented as polar curves. The curve for a
standard cardioid microphone, the most common
unidirectional microphone, can be represented as
r 0.5 0.5sin(?) at its target design
frequency.24 The pattern shifts toward
omnidirectionality at lower frequencies

95
The Golden Ratio

In his book, "The Golden Ratio The Story of Phi,
the World's Most Astonishing Number" (Broadway
Books, 2002), Livio describes among other things
the remarkable connection between avian flight
patterns, stormy weather and cosmic pinwheels.

Livio said the logarithmic spiral is a key shape
for anything that grows, because with growth the
ratio does not change. But logarithmic spirals
appear in totally unrelated phenomena.
"They also appear, interestingly enough, when a
falcon dives toward its prey," Livio said. The
flight pattern allows the bird to maintain a
constant angle. Head cocked, its eyes never
waver. "It allows the falcon to keep its prey
continuously in sight."

Phi (not pi) is the number 1.618 followed by an
infinite string. Take a rectangle whose sides
conform to this Golden Ratio, carve from it a
square, and the remaining rectangle still follows
the ratio.
The Golden Ratio also describes the
ever-expanding nature of what is termed a
logarithmic spiral, not to be confused with the
boring spiral created by a roll of toilet paper.
You've seen the logarithmic spiral in a familiar
seashell belonging to a creature called the
chambered nautilus.

98
Connection to spherical and cylindrical
coordinates

The polar coordinate system is extended into
three dimensions with two different coordinate
systems, the cylindrical and spherical coordinate
systems.

99
3-D (Spherical) Polar Coordinates

The 3-D polar coordinate system or the spherical
coordinate system involves the distance from the
origin and 2 angles (Figure 3). The position of
point P is described as (r, ø,?), where r the
distance from the origin (O), ø the horizontal
azimuth angle measured on the XY plane from the X
axis in the counterclockwise direction, and ?
the azimuth angle measured from the Z axis.
Again, the coordinates are not the same kind.