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)
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
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