Polar Coordinates System

The polar coordinate system is a two-dimensional coordinate system that employs distance and angle to represent points on a plane. It’s similar to a regular coordinate system, but instead of using x and y coordinates, it uses:

1. Radius (r): The distance from a fixed reference point, known as the origin or pole.
2. Angular coordinate (θ): The angle measured counterclockwise from a fixed direction, referred to as the polar axis.

Key features of the polar coordinate system:

1. Points are identified with an ordered pair (r, θ). An example would be the point (2, π/3), meaning it lies 2 units away from the origin while maintaining an angle of π/3 (or approximately 60 degrees) from the polar axis.
2. The angle θ ranges between 0 and 2π (360 degrees). However, negative angles are valid too; these simply imply moving counterclockwise past the polar axis.
3. Variable ‘r’, representing the radius, accepts non-negative values only. When r equals 0, it indicates that the point sits right on top of the origin.

In this article, we will discuss the Polar Coordinate System in detail, including its Properties, Graph, Formula, and Examples.

Polar Coordinate System

In contrast to Cartesian coordinates, the polar coordinate system expresses points in terms of their distance from a central point (the pole) and the angle formed with a reference axis (usually the positive x-axis). It is used to define points uniquely in a plane using two values: the radial distance from a fixed point (called the pole or origin) and the angle from a fixed direction.

Polar coordinates are a way to represent points in a plane using a distance from a reference point (called the pole) and an angle from a reference direction (often the positive x-axis).

In polar coordinates, a point is denoted as (r, θ) where ‘r’ represents the distance from the origin, and ‘θ’ represents the angle from positive x-axis.

Graph of Polar Coordinates

Visualizing polar coordinates involves plotting points in a polar coordinate plane, typically a circle with radii representing ‘r’ and angles dictating the direction.

When constructing of graph related to polar co-ordinates, the knowledge of angles is important as the plotting of the angles is done for representation of co-ordinates.

Below in the graph: Angles π ∕2, π ∕3, π ∕6, 2π , 11π ∕6, 5π ∕3, 3π ∕2, 4π ∕3, 7π ∕6, π , 5π ∕6 and 2π ∕3 are represented for understanding of polar representation.

Polar Coordinates Formula

In polar coordinates, a point is represented by (r, θ), where ‘r’ is the distance from the origin (pole), and ‘θ’ is the angle formed with the reference direction (usually the positive x-axis).

Given:

• x represents the horizontal distance on x-axis,
• y represents the vertical distance on x-axis,
• r is radial distance, and
• θ is angle from positIve x-axis.

Then polar co-ordinates are :

r = √(x² + y²)

θ = tan(y/x)

3D Polar Coordinates

3D polar coordinates are a way to represent points in three-dimensional space using a different coordinate system than the familiar Cartesian coordinates (x, y, z). In 3D polar coordinates, you describe a point’s position using its distance from the origin, an angle θ that represents its azimuthal angle (the angle in the xy plane), and an angle φ that represents its polar angle (the angle from the positive z-axis).

In 3D Polar Coordinates, we need three parameters i.e., ρ, θ and φ.

Symbol	Description
ρ (rho)	Radial distance from the origin to the point.
θ (theta)	Azimuthal angle, measured in the xy plane from the positive x-axis counterclockwise to the point.
φ (phi)	Polar angle, measured from the positive z-axis to the vector connecting the origin and the point.

Conversion Formula for 3D Polar Coordinates

The conversion between Cartesian coordinates (x, y, z) and 3D polar coordinates (ρ, θ, φ) is as follows:

• ρ = √(x² + y² + z²)
• θ = tan^-1(y/x) (the arctangent of y and x, giving the azimuthal angle)
• φ = cos^-1(z / ρ) (the arccosine of the z-coordinate divided by ρ, giving the polar angle)

Conversely, to convert from 3D polar coordinates to Cartesian coordinates:

• x = ρ × sin(φ) × cos(θ)
• y = ρ × sin(φ) × sin(θ)
• z = ρ × cos(φ)

Convert Cartesian Coordinates to Polar Coordinates

When calculating polar coordinates, the conversion from Cartesian coordinates involves the use of trigonometric functions. The distance (r) is determined by the square root of the sum of the squared x and y coordinates, while the angle (θ) is found using the tangent function.

Converting Cartesian coordinates (x, y) to polar coordinates involves determining the radial distance (r) and the angle (θ). The conversion is performed using the following formulas:

Radial distance: The radial distance is the straight-line distance from the origin (0,0) to the point (x, y).

r = √(x² + y²)

Angle (θ): The angle θ is the tangent of the ratio of the y-coordinate to the x-coordinate. Note that special attention should be given to the quadrant in which the point lies to obtain the correct angle.

θ = tan(y/x)

Example: Covert the point P(3, 4) from Cartesian coordinate system to polar coordinates.

Solution:

Step 1: Identify the coordinates of the point in the Cartesian system.

• x = 3
• y = 4

Step 2: Use the following formulas to convert Cartesian to Polar Coordinates:

r = √(x² + y²), and

θ = tan(y/x)

Step 3: Substitute the values of x and y into the formulas:

r = √(x² + y²) = √(3² + 4²) = √(9 + 16) = √25 = 5, and

θ = tan(4/3)

Step 4: Calculate the angle in degrees using a calculator:

≈ 53.13°

Step 5: Write the Polar Coordinates:

(P(3, 4) in Cartesian Coordinates is equivalent to P(5, 53.13°) in Polar Coordinates.

Therefore, the Polar Coordinates of the point P(3, 4) are P(5, 53.13°).

Plotting Points in Polar Coordinates

Graph is plotted for (r, θ) for plotting Points in Polar Coordinates

For polar coordinates (r = 5, θ = 45°)

Polar co-ordinates are plotted as below:

Convert Polar Coordinates to Cartesian Coordinates

Converting polar coordinates to Cartesian coordinates involves transforming the radial distance (r) and angle (θ) back to rectangular form. The conversion is achieved through the following formulas:

Horizontal Coordinate (x)

• The horizontal coordinate is determined by multiplying the radial distance (r) by the cosine of the angle (θ).
• Conversion from polar to Cartesian coordinates along the x-axis is represented as below:

x = r ⋅ cos θ

Vertical Coordinate (y)

• The vertical coordinate is obtained by multiplying the radial distance (r) by the sine of the angle (θ).
• Conversion from polar to Cartesian coordinates along the y-axis is represented as below:

y = r ⋅ sin θ

Example: Covert point (5, 45°) in cartesian coordinate system.

Solution:

Cartesian coordinates using polar coordinates is given by

• x = r × cos(θ)
• y = r × sin(θ)

As cosine of 45° is 1/√2 OR √2/2 and the sine of 45° is 1/√2 OR √2/2. So, the Cartesian coordinates can be written directly as:

• x = 5 × √2 / 2 = 5√2 / 2 ≈ 3.54
• y = 5 × √2 / 2 = 5√2 / 2 ≈ 3.54

The Cartesian coordinates can be calculated and represented as (3.54, 3.54).

Related :

• Polar Representation of Complex Numbers
• Area Between Polar Curves
• Polar and Exponential Forms of Complex Numbers

Difference between Polar and Cartesian Coordinates

Polar and Cartesian Coordinates are two different coordinate system to represent various different point in 2D space. Key differences between these coordinates are:

Polar vs Cartesian Coordinates
Aspect	Cartesian Coordinates	Polar Coordinates
Dimension	Can be used in 2D and 3D (x, y, z)	Primarily used in 2D (r, θ)
Representation	Points are located by their distance from two perpendicular axes (x and y)	Points are located by their distance from a fixed origin (r) and angle from a fixed reference direction (θ)
Visualization	Grid of horizontal and vertical lines	Origin with concentric circles radiating outward and lines at regular angular intervals
Strengths	Simple and intuitive for rectangular shapes and straight lines	Efficient for representing circular and radial relationships
Weaknesses	Can be cumbersome for circular shapes and angles	Can have ambiguity for certain points (e.g., negative radii)
Applications	Plotting points, calculating distances and areas, linear equations, motion in straight lines	Describing circular motion, planetary orbits, wave patterns, polar plots
Example	Plotting the location of a house on a map (x, y)	Describing the position of a planet around the sun (r, θ)

Applications of Polar Coordinate System

The polar coordinate system is useful in many applications, including:

• Navigation: Sailors and pilots often use polar coordinates to specify their location and direction of travel.
• Physics: Polar coordinates are useful for studying circular motion and other problems that involve radial distances and angles.
• Graphics: Polar coordinates are used to create some types of computer graphics, such as spirals and roses.

Advantages of Polar Coordinate System

The polar coordinate system offers several advantages over the traditional Cartesian system, especially in specific situations. Here are some key benefits:

Simplicity for Radial Phenomena:

• Circular and Rotational Motion: Equations describing circular motion become significantly simpler in polar coordinates. Instead of dealing with separate x and y components, you only need the radius and angular coordinate, greatly simplifying analysis of speed, acceleration, and force.
• Radius-Based Descriptions: Objects with radial symmetry, like circles, spirals, and rose curves, have elegant and concise equations in polar coordinates compared to their complex counterparts in Cartesian.

Intuitive Representation:

• Direction and Distance Combine: Polar coordinates naturally capture both direction and distance information in a single unit, making it easier to visualize and reason about problems involving angles and radii.
• Rotating Objects: Situations involving objects rotating around a fixed point are naturally described in polar coordinates, as the angle directly captures the rotational aspect.

Calculus Applications:

• Integration: Calculating areas or volumes of specific shapes, especially those with radial symmetry, can be significantly easier in polar coordinates due to the simpler change-of-variables involved.
• Complex Numbers: Polar coordinates provide an alternative and intuitive way to visualize and manipulate complex numbers, offering geometric interpretations for their operations.

Disadvantages of Polar Coordinate System

While the polar coordinate system offers valuable advantages, it also comes with some limitations and disadvantages. Here are some key points to consider:

Less Intuitive for General Use:

• Non-linear Operations: Unlike Cartesian coordinates where simple addition and subtraction suffice, operations like scaling or translating shapes in polar coordinates involve trigonometric functions, making them less intuitive for basic manipulations.
• Visualizing Distance Relationships: While angles are easy to grasp, distances represented by radii don’t offer the same direct visual understanding as Cartesian x and y values.

Multiple Representations:

• Ambiguity with Angles: The same point can be represented by multiple pairs of angle and radius combinations (e.g., (1, 0°) and (1, 360°)). This can lead to confusion and errors if not handled carefully.
• Negative Radii: Negative radii don’t have a real-world interpretation, introducing unnecessary complexity and potential confusion.

Limited Scope:

• Two-Dimensional: Polar coordinates are restricted to representing points and shapes on a plane. They cannot directly handle three-dimensional situations.
• Complex Shapes: Representing non-radially symmetric or irregular shapes can be challenging and cumbersome in polar coordinates, often requiring conversion to Cartesian for easier manipulation.

Computational Considerations:

• Conversion overhead: Switching between polar and Cartesian systems can involve trigonometric calculations, adding computational overhead, especially for complex problems.
• Specialised Algorithms: Some mathematical operations and numerical methods like gradient descent might require specialised algorithms adapted for polar coordinates.

Solved Questions on Polar Coordinates

Question 1: Convert Cartesian to Polar Coordinates: Given (x, y) = (3, 4), find the polar coordinates (r, θ).

Solution:

Given: (x, y) = (3, 4)

Thus, x = 3 and y = 4

Using formula r = √{x² + y²} and θ = tan^-1(y/x), we get

r = √{3² + 4²} = 5, and

θ = tan^-1(4/3)

Question 2: Convert Cartesian to Polar Coordinates: Given (x, y) = (6, 8), find the polar coordinates (r, θ).

Solution:

Given: (x, y) = (6, 8)

Thus, x = 6 and y = 8

Using formula r = √{x² + y²} and θ = tan^-1(y/x), we get

r = √{6² + 8²} = 10, and

θ = tan^-1(8/6)

Question 3: Convert Polar to Cartesian Coordinates: Given (r, θ) = (2, π /4), find the Cartesian coordinates (x,y).

Solution:

Given: (r, θ) = (2, π /4)

Thus, x = 2 and y = π /4

Using formula x = rcos(θ) and y = rsin(θ)

x = 2cos( π /4)

y = 2sin( π /4)

Question 4: Convert Polar to Cartesian Coordinates: Given (r, θ) = (6, π/3), find the Cartesian coordinates (x,y).

Solution:

Given: (r, θ) = (6, π /3)

Thus, x = 6 and y = π /3

Using formula x = rcos(θ) and y = rsin(θ)

x = 6cos( π /3)

y = 6sin( π /3)

Question 5: Convert Cartesian to Polar Coordinates: Given (x, y) = (8, 15), find the polar coordinates (r, θ).

Solution:

Given: (x, y) = (8, 15)

Thus, x = 8 and y = 15

Using formula r = √{r² + y²} and θ = tan^-1(y/x), we get

r = √{8² + 15²} = 17, and

θ = tan^-1(15/8)

Practice Problems on Polar Coordinates

Problem 1: Convert Cartesian to Polar Coordinates: Given (x, y) = (8, 6), find the polar coordinates (r, θ).

Problem 2: Convert Cartesian to Polar Coordinates: Given (x, y) = (13, 15), find the polar coordinates (r, θ).

Problem 3: Convert Polar to Cartesian Coordinates: Given (r, θ) = (5, π /8), find the Cartesian coordinates (x,y).

Problem 4: Convert Polar to Cartesian Coordinates: Given (r, θ) = (4, π /4), find the Cartesian coordinates (x,y).

Problem 5: Determine the distance between the points (9, 6) and (3, 2).

Frequently Asked Questions on Polar Coordinates

What are Polar Coordinates?

Polar coordinates represent the location of a point based on its distance from a reference point (the origin) and the angle from a reference direction (usually the positive x-axis).

What are Rectangular Polar Coordinates?

Rectangular coordinates represent a point in a 2D plane using horizontal (x) and vertical (y) distances from the origin. Polar coordinates, on the other hand, use radial distance (r) and angular measure (θ) from the origin.

What is Curvature in Polar Coordinates?

Curvature in polar coordinates, denoted as κ, measures how sharply a curve bends at a point. It’s calculated using κ = (r × d²θ/ds²) / (dr/ds), considering the polar angle, radial distance, and arc length.

How to Convert Cartesian to Polar Coordinates?

To convert Cartesian coordinates (x, y) to polar coordinates (r, θ), you can use the following formulas:

1. Calculate the radial distance (r): r = √(x²+ y²)
2. Calculate the polar angle (θ): θ = arctan(y / x)

What is Gradient in Polar Coordinates?

In polar coordinates, the gradient is given by ∇f = (∂f/∂r)∙([Tex]\hat{r} [/Tex]) + (1/r)∙(∂f/∂θ)∙([Tex]\hat{\theta} [/Tex]), representing the rate of change of a scalar function in both radial and angular directions.

What is Circle in Polar Coordinates?

In polar coordinates, a circle with its center at the origin (0,0) is described by a simple equation:

r = a

Here, “r” represents the radial distance from the origin to a point on the circle, and “a” is a positive constant representing the radius of the circle.

What are Polar Coordinates of Sphere?

The polar coordinates of a point on a sphere are represented as (r, θ, φ), where:

• r is the radial distance from the origin to the point.
• θ is the polar angle, measured from the positive x-axis in the xy-plane.
• φ is the azimuthal angle, measured from the positive z-axis.

Define Curves in Polar Coordinates.

To define curves in polar coordinates, express the radial distance (r) as a function of the polar angle (θ), typically written as r = f(θ). Plot the resulting points to visualize the curve.

What is Cauchy Riemann Equations in Polar Coordinates?

Cauchy-Riemann equations in polar coordinates are given by:

• ∂u/∂r = (1/r) ∂v/∂θ
• (1/r) ∂u/∂θ = -∂v/∂r