final ppt of tracing of polar curves
DESCRIPTION
presentationTRANSCRIPT
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Tracing of Curves For
POLAR COORDINATES
-
Rootvesh Mehta
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A coordinate system represents
a point in the plane by an ordered
pair of numbers called coordinates.
Usually, we use Cartesian coordinates,
which are directed distances from two
perpendicular axes.
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Here, we describe a coordinate system
introduced by Newton, called the polar
coordinate system.
It is more convenient for many purposes.
POLAR COORDINATES
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Polar Coordinates &
Tracing of Polar Curves
POLAR COORDINATES
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POLE
We choose a point in the plane
that is called the pole (or origin)
and is labeled O.
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POLAR AXIS
Then, we draw a ray (half-line) starting
at O called the polar axis(Initial Line).
This axis is usually drawn horizontally to the right
corresponding to the positive x-axis in Cartesian
coordinates.
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ANOTHER POINT
If P is any other point in the plane, let:
r be the distance from O to P.
θ be the angle (usually measured in radians)
between the polar axis and the line OP.
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POLAR COORDINATES
P is represented by the ordered pair (r, θ).
r, θ are called polar coordinates of P.
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In rectangular coordinates:
•We break up the plane into a grid of
horizontal and vertical line lines.
•We locate a point by identifying it as
the intersection of a vertical and a
horizontal line.
In polar coordinates:
•We break up the plane with circles
centered at the origin and with rays
emanating from the origin.
•We locate a point as the intersection
of a circle and a ray.
Coordinate systems are used to locate the position of a point.
(3,1) (1,/6)
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POLAR COORDINATES
We use the convention that
an angle is:
Positive—if measured in the counterclockwise
direction from the polar axis.
Negative—if measured in the clockwise
direction from the polar axis.
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If P = O, then r = 0, and we agree that
(0, θ) represents the pole for any value
of θ.
POLAR COORDINATES
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We extend the meaning of polar
coordinates (r, θ) to the case in which
r is negative—as follows.
POLAR COORDINATES
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POLAR COORDINATES
We agree that, as shown, the points (–r, θ)
and (r, θ) lie on the same line through O
and at the same distance | r | from O, but
on opposite sides of O.
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POLAR COORDINATES
If r > 0, the point (r, θ) lies in the same
quadrant as θ.
If r < 0, it lies in the quadrant on the opposite
side of the pole.
Notice that (–r, θ)
represents
the same point
as (r, θ + π).
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POLAR COORDINATES
Plot the points whose polar coordinates
are given.
a. (1, 5π/4)
b. (2, 3π)
c. (2, –2π/3)
d. (–3, 3π/4)
Example 1
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POLAR COORDINATES
The point (1, 5π/4) is plotted here.
Example 1 a
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The point (2, 3π) is plotted.
Example 1 bPOLAR COORDINATES
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POLAR COORDINATES
The point (2, –2π/3) is plotted.
Example 1 c
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POLAR COORDINATES
The point (–3, 3π/4) is plotted.
It is is located three
units from the pole
in the fourth quadrant.
This is because
the angle 3π/4 is in
the second quadrant
and r = -3 is negative.
Example 1 d
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CARTESIAN VS. POLAR COORDINATES
In the Cartesian coordinate system, every
point has only one representation.
However, in the polar coordinate system,
each point has many representations.
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CARTESIAN VS. POLAR COORDINATES
For instance, the point (1, 5π/4) in
Example 1 a could be written as:
(1, –3π/4), (1, 13π/4), or (–1, π/4).
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CARTESIAN & POLAR COORDINATES
In fact, as a complete counterclockwise
rotation is given by an angle 2π, the point
represented by polar coordinates (r, θ) is
also represented by
(r, θ + 2nπ) and (-r, θ + (2n + 1)π)
where n is any integer.
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Unit Circle
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CARTESIAN & POLAR COORDINATES
The connection between polar and Cartesian
coordinates can be seen here.
The pole corresponds to the origin.
The polar axis coincides with the positive x-axis.
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CARTESIAN & POLAR COORDINATES
If the point P has Cartesian coordinates (x, y)
and polar coordinates (r, θ), then, from
the figure, we have:cos sin
x y
r r
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cos
sin
x r
y r
Equations 1
Therefore,
CARTESIAN & POLAR COORDS.
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CARTESIAN & POLAR COORDS.
Although Equations 1 were deduced from
the figure (which illustrates the case where
r > 0 and 0 < θ < π/2), these equations are
valid for all values of r and θ.
See the general
definition of sin θ
and cos θ
in Appendix D.
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CARTESIAN & POLAR COORDS.
Equations 1 allow us to find
the Cartesian coordinates of a point
when the polar coordinates are known.
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CARTESIAN & POLAR COORDS.
To find r and θ when x and y are known,
we use the equations
These can be
deduced from
Equations 1 or
simply read from
the figure.
2 2 2 tany
r x yx
Equations 2
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CARTESIAN & POLAR COORDS.
Convert the point (2, π/3) from polar to
Cartesian coordinates.
Since r = 2 and θ = π/3,
Equations 1 give:
Thus, the point is (1, ) in Cartesian coordinates.
Example 2
1cos 2cos 2 1
3 2
3sin 2sin 2. 3
3 2
x r
y r
3
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CARTESIAN & POLAR COORDS.
Represent the point with Cartesian
coordinates (1, –1) in terms of polar
coordinates.
Example 3
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CARTESIAN & POLAR COORDS.
If we choose r to be positive, then
Equations 2 give:
As the point (1, –1) lies in the fourth quadrant,
we can choose θ = –π/4 or θ = 7π/4.
Example 3
2 2 2 21 ( 1) 2
tan 1
r x y
y
x
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CARTESIAN & POLAR COORDS.
Thus, one possible answer is:
( , –π/4)
Another possible answer is:
( , 7π/4)
Example 3
2
2
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CARTESIAN & POLAR COORDS.
Equations 2 do not uniquely determine θ
when x and y are given.
This is because, as θ increases through the interval
0 ≤ θ ≤ 2π, each value of tan θ occurs twice.
Note
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CARTESIAN & POLAR COORDS.
So, in converting from Cartesian to polar
coordinates, it’s not good enough just to find r
and θ that satisfy Equations 2.
As in Example 3, we must choose θ so that
the point (r, θ) lies in the correct quadrant.
Note
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POLAR CURVES
The graph of a polar equation r = f(θ)
[or, more generally, F(r, θ) = 0] consists
of all points that have at least one polar
representation (r, θ), whose coordinates
satisfy the equation.
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POLAR CURVES
What curve is represented by
the polar equation r = 2 ?
The curve consists of all points (r, θ)
with r = 2.
r represents the distance from the point
to the pole.
Example 4
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POLAR CURVES
Thus, the curve r = 2 represents
the circle with center O and radius 2.
Example 4
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POLAR CURVES
In general, the equation r = a represents
a circle O with center and radius |a|.
Example 4
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POLAR CURVES
Sketch the polar curve θ = 1.
This curve consists of all points (r, θ)
such that the polar angle θ is 1 radian.
Example 5
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POLAR CURVES
It is the straight line that passes through O
and makes an angle of 1 radian with the polar
axis.
Example 5
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POLAR CURVES
Notice that:
The points (r, 1) on
the line with r > 0 are
in the first quadrant.
The points (r, 1) on
the line with r < 0 are
in the third quadrant.
Example 5
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POLAR CURVES
a. Sketch the curve with polar equation
r = 2 cos θ.
b. Find a Cartesian equation for this curve.
Example 6
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POLAR CURVES
First, we find the values of r for
some convenient values of θ.
Example 6 a
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POLAR CURVES
We plot the corresponding points (r, θ).
Then, we join these points
to sketch the curve—as
follows.
Example 6 a
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The curve appears to be a circle.
Example 6 aPOLAR CURVES
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POLAR CURVES
We have used only values of θ between 0
and π—since, if we let θ increase beyond π,
we obtain the same points again.
Example 6 a
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POLAR CURVES
To convert the given equation to a Cartesian
equation, we use Equations 1 and 2.
From x = r cos θ, we have cos θ = x/r.
So, the equation r = 2 cos θ becomes r = 2x/r.
This gives:
2x = r2 = x2 + y2 or x2 + y2 – 2x = 0
Example 6 b
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POLAR CURVES
Completing the square,
we obtain:
(x – 1)2 + y2 = 1
The equation is of a circle with center (1, 0)
and radius 1.
Example 6 b
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sin1r
2
211
3
87.1
2
31
6
5.1
2
11
0 101
6
2
1
2
11
3
13.0
2
31
2
011
This type of graph is called a cardioid.
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Tracing of Polar curves
It is important to consider following properties for tracing of polar curves:-
Symmetry
Curve Passes through the pole and tangent at the pole
Points of intersection
Loop of the curve
Region or Extent of the curve
Increasing/Decreasing nature of curve
Direction of Tangent
Asymptote
Closedness of the Curve
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1) SYMMETRY
A) Symmetry of Curve about Initial Line-If a polar equation is unchanged
when θ is replaced by –θ, the curve is symmetric about the polar
axis(Initial Line).
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1) Symmetry
B) Symmetry about Pole:-If the equation is unchanged when r is replaced
by –r, or when θ is replaced by θ + π, the curve is symmetric about the
pole.
This means that
the curve remains
unchanged if we rotate
it through 180° about
the origin.
Example:-
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1) Symmetry
C) Symmetry about line θ = π/2 (Normal Line):- If the equation is
unchanged when θ is replaced by π – θ, the curve is symmetric about the
vertical line θ = π/2.
Example:-
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1)Symmetry
D) Symmetry about line θ = π/4 :-
Example:-
If the equation is unchanged when θ is replaced by π/2 – θ, the
curve is symmetric about the vertical line θ = π/4.
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1) Symmetry
E) Symmetry about line θ = 3π/4 :-
Example:-
If the equation is unchanged when θ is replaced by 3π/2 – θ, the
curve is symmetric about the vertical line θ = 3π/4.
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2) Curve Passes through the pole and tangent to the curve at the pole:-
Example:-
If r=0 for some value of say = then the curve is passes through the pole and that
= is the tangent to the curve at the pole
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3) Point of Intersection
Example:-
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4) Loop of the curve
If the curve intersect initial line at two places A and B and curve is
symmetric about initial line then there exists loop of the curve between A
and B.
Example:-
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5) Region or Extent of the curve
1) Region where curve exists :-Find the least value a and gretest
value b of rsuchthat a < r < bthen the curve lies in the region between
two circles of radius a and b
2
2) Region where curve does not exists :-Find the value of for which
r 0 i.e.r is imaginaryand in such a region curve does not exists.
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6) Direction of the tangent
Here
At any point (r, ) the tangent is obtained by
tan = r where is the angle between radial line and tangent.d
dr
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7) Increasing and decreasing nature of the curve
If 0 then r is increasing and If 0 then r is decreasing with
increasing value of
dr dr
d d
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8) Closedness of the Curve
lt r=f( ) be the polar equation of the curve . If r=f( )=f(2 + )
then the given curve is closed.
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9) Asymptote
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Conclusion:-
So you need to verify above properties for
tracing of curve though you need to do many
time plotting and need to draw curve.
Thank you