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Polar Coordinates

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 Sub Topics Polar coordinate is a method of presentation of points in the plane with use of Ordered Pair. This polar coordinate system comprises of an origin, pole, a Ray of specific angle and a polar axis. The polar axis is any line which initiates from the origin and extends to the indefinite Point in any of the prescribed direction. The Position of the point can be determined by the position and distance from the origin and by its angle. If the rotation of polar axis is anticlockwise then the angle generated will be positive. If it is clockwise the angle generated will be negative. These are useful in solving circular Symmetry. Besides Cartesian coordinate system the polar coordinate is another way to find the position of the point. Polar coordinates help us to know about the following aspects: How to plot a point when its polar coordinate is given. How polar coordinates can be converted in Cartesian coordinates. Again conversion of Cartesian into polar coordinate. Identifying the angle whether they are positive or negative. Whether equation is satisfying the curve or not. The polar coordinate system is a two dimensional coordinate system which is ascertained by distance from any fixed point and from the angle in the fixed direction. The fixed point in the graph is a pole; the distance which is measured from the pole is its radius. The equation which is defined in the form of algebraic curve given in the polar coordinates is polar equation. As the polar coordinate system has the circular nature so many curves can be explained by it. The polar rose, Archimedean spiral, Lemniscate, Limacon and cardioid are among its best renowned curves.

Locus and Polar Coordinates

Locus for a Point can be defined as a curve or a path that results from the condition (s) that is satisfied by the point for an Algebraic Relation governed by some fixed rule. This statement is valid only for those points which are lying in the plane and not in the outer region of the plane, i.e., the equation of loci cannot be found for those points lying in the exterior.

And as we know that a point can be located or represented in a plane using two forms of systems namely the Rectangular/Cartesian Coordinate System or the Polar Coordinate System.

A proper relation can be framed between the locus and the Polar Coordinates by using these coordinates to find the locus of any arbitrary point lying in the plane.
The simple technique which can be followed for doing this is by substituting the values of polar coordinates in place of x and y, i.e.,

x = h cos A
y = h sin A

The steps for finding the equation of the locus of a point P being the same:
First of all assign the coordinates to the point of which the locus has to be found, i.e. P (n, m).
The second thing which we need to remember is to express the given conditions as equations in terms of the known quantities and unknown parameters.
The unknown parameters should be eliminated, such that only the know quantities are left for consideration.
Replace n by x, and m by y. The resulting equation representing the locus of the point P can be obtained by replacing n by x, and m by y.

All the points lying in the plane and making the same angle with the positive x- axis will have the same Slope and also the equation of the loci.

Applications of Polar Coordinates

The Polar Coordinates are defined as two-dimensional co-ordinate system, used to locate arbitrary Point positions lying on the circumference of a single two-dimensional plane like a Circle and situated at a particular distance from a fixed point.
Their use in this context is where the phenomenon being considered is essentially related to the direction and the length from a Centre point of the plane.
Moreover, many physical systems—such as those concerned with objects moving around a central point or with phenomena originating from a central point—are simpler and more spontaneous to model using polar coordinates.

The initial incentive taken in the field of explaining the applications of Polar Coordinates was the study related to circular and orbital motion.
Navigation can also be a noticeable application of Polar coordinates, as we consider the direction of travel can be given as an angle and distance from the object being considered. Thus the destination can be come to know by knowing the navigation direction of the Polar coordinates.

According to the conventions that are followed by the aircrafts using the polar coordinates the magnetic north can be understood as moving in a direction along 360 and the rest directions are specified by 90, 180 and 270 respectively.
Another application of Polar Coordinate Systems is modelling which can be used to display radial Symmetry providing natural settings for the polar coordinate system, with the central point acting as the pole.
For Example we can consider the following:
Groundwater flow equation applied to radically symmetric wells.
Systems including gravitational fields with a radical force also have a good usage of the polar coordinate system.
Modelling with polar coordinates also cover their usage in asymmetric systems like we can get a proportionate response to an incoming sound signal from a given direction is usually found in microphone's pickup pattern.

Transformation from Rectangular to Polar Coordinates

A plane can be defined as a point having zero-dimensions or a line having one-dimension or a space which encompasses three-dimensions. Any Point taken on this plane can be represented in two different ways that are:
1. Rectangular coordinates and
2. Polar coordinates.
The Rectangular coordinates are present in the form of P(x, y), where 'x' and 'y' are the horizontal and vertical distances from the origin.
The Polar coordinate system is defined in terms of distance from a fixed point and an angle when viewed from a particular direction. Let the distance of a point P(x, y) from origin (an arbitrary fixed point) be denoted by ‘h’ denoted by the symbol O). Consider the angle between the radial line from the point P to O and the given line “θ = 0” (a kind of positive axis for our polar coordinate system be angle ‘A’. Where,
h ≥ 0 & 0≤ A < 2π.
The transformation of Rectangular Co-ordinates to the Polar co-ordinates can be done using certain formulae:

In the given diagram we have,
By the rule of Pythagoras:
r = √ (x2 + y2)
And the Slope can be found by tan A,
or tan A = y / x , so therefore:
A = tan-1 (y / x)
So the rectangular point: (x, y) can be converted to polar coordinates like this:
(√ (x2 + y2), tan-1 (y / x)).
An Example can be taken to understand this transformation better,
Example: A point is having rectangular coordinates as (3, 4). Find out the corresponding Polar Coordinates.
Solution: r = radius or distance of the point from the Centre (fixed point) = square root of
(32 + 42) = 5, and
Angle A = tan-1(4/3) = 53.13º
so, the polar co-ordinates can be given as
(r, A) = (5, 53.13º).

Equation of a Locus

Locus for a Point can be defined as a curve or a path that results from the condition (s) that is satisfied by the point for an Algebraic Relation governed by some fixed rule. This statement is valid only for those points which are lying in the plane and not in the outer region of the plane, i.e., the equation of loci cannot be found for those points lying in the exterior.

And as we know that a point can be located or represented in a different way we can say that if a point moves in a plane following certain geometrical conditions, it traces out a path. This path of the moving point is called its locus.

To explain the geometrical meaning of locus we need to derive an equation which can be described as a relation existing between the coordinates of all the point on the path, and which holds for no other points except those lying on the path.

If we are interested in finding the equation of the locus of a point we need to follow these steps:

The first step is to locate the coordinates of the point for which the locus has to be found, i.e. P (h, k). This helps you identifying the location of the point lying in the plane for measurements.

Next step is to determine the known quantities and unknown parameters while expressing the given conditions as equations in terms of the known quantities and unknown parameters. This assists you for further calculation.

Unknown parameters should be omitted.

Replacing the point h by ‘x’ and k by ‘y’, results into an equation representing the locus of the point P.

All the points making the same angle with the real axis will have the same Slope and also the equation of the loci.

Polar Coordinates to Cartesian

Coordinates are of two types; one is Cartesian coordinates and second is Polar Coordinates. We can convert polar coordinates to Cartesian coordinates and from Cartesian coordinates to polar coordinates.
Basically we use graph to mark these coordinates. Cartesian coordinates consist of two points, position on x- axis and Position of y- axis. These coordinates define how far an object is from origin. Polar coordinates also have coordinates as x and y- axis, which defines the position of the object on the graph but in addition it also has angles defined. This angle lies between x- axis and y- axis.
Four points are required to convert the coordinates from one form to another. These are position of x and y- axis, hypotenuse and angle which lie between the x and y- axis.
Let us see how we can convert polar coordinates to Cartesian coordinates: In polar coordinates we are given two points 'r' and angle 'θ'. Here side 'r' is √x2 + y2. When we convert polar coordinates to Cartesian coordinates we need to calculate ‘x’ and ‘y’ points.
In polar form the x- coordinates are associated with cos function and y- coordinates are associate with sin function.
Let us understand the conversion of polar to Cartesian with an example. Assume we have polar coordinates 'r' and 'θ' as (13, 22.60). Now we have to calculate x and y axes.
So cos(22.60) = x / 13 => x = 12.006
This is the position of x- axis on the graph.
Now calculate the y axis: sin (22.60) = y / 13 => 4.996,
This is the position of y- coordinates. Now Cartesian coordinates are (12, 5).

Graphing Polar Coordinates

Polar coordinate system is defined in terms of distance from a fixed Point and an angle when viewed from a particular direction. Let the distance of a point P(X, Y) from origin (an arbitrary fixed point) be denoted by ‘H’ and origin is denoted by the symbol 'O'. Consider the angle between the radial line from point 'P' to 'O' and given line “θ = 0” (a kind of positive axis for our polar coordinate system be angle ‘s’.
Graphing Polar Coordinates means to plot them in Cartesian System via Transformations:
X = H cos s and,
Y = H sin s,
Where, H ≥ 0 & 0≤ s < 2π.

Now each and every point P(X, Y) lying in ordinary x - y plane can be written in this new (h, A)-form. It being a consequence of the fact that 'P' lies on circumference of some Circle which is centered at the origin 'O' and has a radius 'R' where, R = H. That is the distance from point 'P' to origin is equal to that of Radius of Circle. From these above mentioned relationships we find that coordinates of our point 'P' are satisfying the equation: X2 + Y2 = H2,
Substituting the values of 1 as,
cos2 s+ sin2 s = 1,
We get,
X2 + Y2 = H2 (cos2 s+ sin2 s),
⇒X2 + Y2 = H2,
Thus providing proof for point P(X, Y) to lie on circle of radius 'R' centered at 'O'.
Polar coordinates can be explained by taking a suitable Example as follows:
What is (4, 3) in Polar Coordinates?
Solution: H2 = 42 + 32,
H = √ (42 + 32)
H= √ ( 16 + 9) = √ (25) = 5
Use the Tangent Function to find the angle:
tan ( s ) = 3 / 4,
S = tan-1 (3 / 14).

Polar Coordinates Integration

Polar Coordinates Integration Calculus can be used to calculate the length of the arc defined by in Polar Coordinates form (r and Ө) for any curve r (Ө). Let “L” represent the length of the arc along the curve between to ends P and Q. Where, these points correspond to“Ө= a” and “Ө= b” such that their difference lies between 0 and 2∏.

Length of arc can be given by following Definite Integral (limit “a” to “b”):
L = ∫((r (Ө)) 2 + [dr (Ө) / d Ө] 2 d Ө) ½,
Region of integration “R” covered under this arc AB is bounded by the curve (Ө)and lines “Ө = a” and “Ө = b”. Area of this region can be given as definite integral (limit “a” to “b”):
R = 1/2 ∫ r (Ө)) 2 d Ө,
Cartesian coordinates can be used to calculate a minute area element as: d A = d x d y. For converting Cartesian coordinates to polar form we use Jacobian determinant given as:
J = det (δ (x, y) / δ (r, Ө)) = r cos2 Ө + r sin2 Ө = r (cos2 Ө + sin2 Ө) = r,
Thus, area of element in polar form can be written as:
d A= d x d y = J dr d Ө = r dr d Ө,
So, integration of a function given in form of polar form can be given with limits of first integral as 0 to r (Ө) and that of second as 'a' to 'b':
∫∫R f (r, Ө) d A = a∫b 0∫r f(r, Ө) r dr d Ө,
Thus it is possible to represent area of any region 'R', covered by some curve r (Ө) in form of polar coordinates integration also.