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:

Calculate the radial distance (r): r = √(x²+ y²)
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

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:

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

Key features of the polar coordinate system:

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.
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.
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.

Table of Content

What is Polar Coordinate System?
What are Polar Coordinates?
Graph of Polar Coordinates
Polar Coordinates Formula
Cartesian to Polar Coordinates Conversion
Polar to Cartesian Coordinates Conversion