Transcript Polar coordinate system

Polar Coordinate System
CALCULUS-III
Dr. Farhana Shaheen
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.
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 θ is the azimuth angle measured
from the polar axis in the counterclockwise
direction. Thus, the position of point P is
described as (r, θ ). Here r & θ are the 2-D polar
coordinates.
Figure: 1
 Any
point P in the plane has its position in
the polar coordinate system determined by
(r, θ).
Some Points With Their Polar
Coordinates
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.
A point in Cartesian Plane
Polar coordinates
coordinates are in the form (r, θ),
where 'r' is the distance from the origin to
the point, and θ is the angle measured
from the positive 'x' axis to the point:
 Polar
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:
The relationship between Polar and Cartesian
coordinates
x = r Cos θ, y = r Sin θ

1. Polar to Rectangular
 From the diagram above, these formulas convert polar
coordinates to rectangular coordinates:
 x = r cos θ, y = r sin θ.
 So the polar point (r, θ) can be converted to
rectangular coordinates as:
 (x, y) = ( r cos θ, r sin θ)
 Example: A point has polar coordinates:
 (5, 30º). Convert to rectangular coordinates.
 Solution: (x, y) = (5cos30º, 5sin30º)

= (4.3301, 2.5)
Converting between polar and
Cartesian coordinates

The two polar coordinates r and θ can be
converted to the Cartesian coordinates x and y
by using the trigonometric functions sine and
cosine:

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while the two Cartesian coordinates x and y can
be converted to polar coordinate r & θ , using the
Pythagorean theorem) as follows:
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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 θ = y/x implies
θ = tan-1( y/x )
So the rectangular point (x,y) can
be converted to polar coordinates
like this:
( r,θ) = ( r, tan-1( y/x ) )
To plot a point in Polar Coordinate
 We
first mark the angles, in the anticlockwise direction from the polar axis.
Negative Distance
OQ is extension of OP
 With
coordinates P(r,θ) and Q(-r, θ+π)
 For

any real r > 0 and for all integers k:
A planimeter, which mechanically
computes polar integrals
 A planimeter
is a measuring instrument
used to determine the area of an arbitrary
two-dimensional shape.
Cartesian equations of Parabolas:
the original graph y=x2 up 2 units.
The resultant graph is y= x2+2
 Move
Polar and Cartesian equations
of a Parabola
Polar and Cartesian equations
of a Parabola
Example:
 Find
the polar equation of each of the
following curves with the given Cartesian
equation:

a)
 B)
x=c
x2 y + y3 = – 4
Solution
To convert Cartesian equation into
polar equation
 Example:
Polar Equations of Straight Lines
θ = α, for any fixed angle α.
Exp: θ = π/4
Straight Lines
 Standard
equation
 of straight line
 in Cartesian coordinates:
 y = mx + c
Polar Equations of Straight Lines
r Cos θ = k; or r = k Sec θ.
 It is a vertical line through k.
 It is equivalent to the Cartesian equation
 x = k.

r Sin θ = k; or r = k Csc θ.
 It is a horizontal line through k.
 It is equivalent to the Cartesian equation
 y = k.

Polar equation of a curve
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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 θ. The
resulting curve then consists of points of the form (r(θ), θ) 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(−θ) = r(θ) the curve will be symmetrical about the
horizontal (0°/180°) ray, if r(π − θ) = r(θ) it will be symmetric about
the vertical (90°/270°) ray, and if r(θ − α°) = r(θ) it will be rotationally
symmetric α° 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.
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(θ) and r= a ± b cos(θ)
 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(nθ) or r= a cos(nθ) for n>1.
 When n is an odd number, the curve has n petals but
when n is even the curve has 2n petals.
Polar Equations of Circles
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r = k : A circle of radius k centered at the origin.
r = a sin θ : A circle of radius |a|, passing through
the origin. If a > 0, the circle will be
symmetric about the positive y-axis; if a < 0, the
circle will be symmetric about the
negative y-axis.
r = a cos θ: A circle of radius |a|, passing through
the origin. If a > 0, the circle will be
symmetric about the positive x-axis; if a < 0, the
circle will be symmetric about the negative
x-axis.
Equations of Circle
A circle with equation r(θ) = 1
 The general equation for a circle with a center at
(r0, φ) and radius a is


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This can be simplified in various ways, to
conform to more specific cases, such as the
equation
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for a circle with a center at the pole and radius a.
A circle with equation r(θ) = 1
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
Graph Polar Equations
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Step 1
Consider r= 4 sin(θ) as an example to learn how to graph polar coordinates.
Step 2
Evaluate the equation for values of (θ) between the interval of 0 and π. Let θ
equal 0, π /6 , π /4, π /3, π /2, 2π /3, 3π /4, 5π /6 and π. 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
θ = π /6. Enter into the calculator 4 sin(π /6). The value for r is 2 and the point
(r, θ) is (2, π /6). Find r for all the (θ) values in Step 2.
Step 4
Plot the resulting (r, θ ) points from Step 3 which are (0,0), (2, π /6), (2.8, π /4),
(3.46,π /3), (4,π /2), (3.46, 2π /3), (2.8, 3π /4), (2, 5π /6), (0, π) 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.
Simplify the Graphing of Polar
Equations

Look for symmetry when graphing these functions.
As an example use the polar equation r=4 sinθ.
 You only need to find values for θ between π (Pi)
because after π the values repeat since the sine
function is symmetrical.
 Step 2
 Choose the values of θ that makes r maximum,
minimum or zero in the equation. In the example
given above r= 4 sin (θ), when θ equals 0 the value
for r is 0. So (r, θ) is (0,0). This is a point of intercept.
 Step 3
 Find other intercept points in a similar manner.
Graphing Polar Equations
 Example
1: Graph the polar equation
given by
r = 4 cos t
 and
identify the graph.
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 = π and r is
equal to zero for t = π/2.
Plotting of points in polar
coordinates
Join the points drawing a smooth
curve r = 4 cos t
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.
Construction of a limacon
Polar Equations of Limacons
 Equations
of limacons have two general
forms:
 r = a ± b sin θ and r = a ± b cos θ:
 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.
Equations of limacon

 If
r = a ± b Cos θ; r = a ± b Sin θ
|a| > |b| then you have a dimple;
 If |a| = |b| then you have a cardioid;
 If |a| < |b| then you have an interior lobe.
Graphs of Limacons
 |a|
>|b|
|a| = |b|
|a| < |b|
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 θ) . Symmetric about the positive y-axis if
`+`; symmetric about the
 negative y-axis if `- '.
 r = a(1 ± cos θ) . 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 2|a| and the
point furthest away from the origin will lie on the axis of
symmetry.
Limacons
Dimpled Limacons
 r=3/2+cos(t)
(purple)
r'=3/2-sin(t) (red)
If |a| < |b| then you have an interior
lobe in Limacon
The family of limaçons is varied
by making a range from -2 to 2,
and then back to -2 again.
Limacon: Pedal curve of a circle
Graph for the equation
r = 2 + 2 sin t (Cardiod)
t
= 0, r = 2
 t = π/6,r = 3.0
 t = π/4,r = 3.4
 t = π/3,r = 3.7
 t = π/2,r = 4
 t = 2π/3,r = 3.7
 t = 3π/4,r = 3.4
 t = π,r = 2
Cardioid
 r=1+cos(t)
 Changing
+ b to - b has the same effect
on the cardiod as with the other limacons;
that is a reflection occurs.
 r=1+cos(t) (magenta)
r=1- cos(t) (purple)
If |a| = |b|, the cardioid will increase
or decrease in size depending on
the value of a and b
 r=0.5+0.5*cos(t)
(black)
r=2+2*cos(t) (purple)
r=3+3*cos(t) (red)
r=4+4*cos(t) (blue)
Rose curves
r
= a Sin nθ;
 r = a Cos nθ;
where n > 1.
Graph has n petals
if n is odd, and 2n
petals if n is even.
Polar rose
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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 nθ;
r = a Cos nθ, for n > 1.
If n is an integer, these equations will produce an npetalled 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.
A polar rose with equation
r(θ) = 2 sin 4θ
Graph for the equation r = 4 cos 2t
t
= 0, r = 4
 t = π/6,r = 2
 t = π/4,r = 0
 t = π/2,r = -4
 t = π/3,r = -2
 t = 2π/3,r = -2
 t = 3π/4,r = 0
 t = π,r = 4
Pretty Petals
Roses
 Consider
the following polar equations
r = cos (2 t) (light red)
r = 3 cos (2 t) (heavy red)
and their associated graphs.
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:
Three-petal flowers
Four- petal flowers
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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.
Five-petal flowers
Spirals
 Logarithmic
Spiral
The logarithmic spiral is a spiral whose
polar equation is given by r =aebθ,
 where r is the distance from the origin, θ 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.
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".
Logarithmic Spiral r = a bθ
 The
distance between successive coils of
a logarithmic spiral is not constant as with
the spirals of Archimedes.

Spirals of Archimedes
 Polar
graphs of the form r = aθ + 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.
Archimedean spiral

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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, θ) it can be
described by the equation
r = aθ + b
with real numbers a and b. Changing the parameter a
will turn the spiral, while b controls the distance between
successive turnings.
The Archimedean spiral
 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.
Hyperbolic spiral
 Any
polar equation that has the form
r = a/θ where a>0 is a hyperbolic spiral.
Cornu Spiral in Complex Plane

A plot in the complex plane of the points B(t)==S(t)+iC(t)
Fermat's Spiral
 Fermat's
spiral, also known as the
parabolic spiral, is an Archimedean spiral
having polar equation r2 = a2θ

Spirals in Nature
 In
nature, you may have noticed that
shells of some sea creatures are shaped
like logarithmic spirals particularly the
nautilus.
Spirals in Nature
Spiral galaxies
Spiral galaxies
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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.
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.
Curves that are close to being
logarithmic spirals
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In several natural phenomena one may find curves that are close to being
logarithmic spirals. Here follows some examples and reasons:
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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.
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Spirals… Down the drain
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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.
Romanesco broccoli
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.
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 curves—such as the Archimedean spiral—whose
equation in the Cartesian coordinate system would be much more
intricate. Moreover, many physical systems—such as those
concerned with bodies moving around a central point or with
phenomena originating from a central point—are 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.
Position and navigation
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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]
Modeling
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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 inversesquare 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(θ)
at its target design frequency.[24] The pattern shifts toward
omnidirectionality at lower frequencies
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 everexpanding 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.
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.
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,θ), 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 θ = the azimuth angle measured from the
Z axis. Again, the coordinates are not the same kind.
Figure 3