# Introduction to Spherical Coordinates

## What is Spherical Coordinate System

The spherical coordinate system is a coordinate system used to locate the position of points in three-dimensional space. The position of any point (P) in spherical coordinates can be specified using the ordered triplet (ρ, θ, φ). Where:

- Radial distance (ρ) is the distance from the origin (O) to point (P).
- Azimuthal angle (θ) is the angle measured from the reference direction (positive x-axis) to the orthogonal projection of the line segment (OP) on the xy-plane.
- Polar angle (φ) is the angle between the positive z-axis and the line segment (OP).

Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates.

### Plotting Points Using Spherical Coordinates Example 1.1

Plot the point with spherical coordinates (3, 40°, 65°).

Step 1) We identify the ρ, θ, and φ coordinates as:

ρ-coordinate = 3 θ-coordinate = 40° φ-coordinate = 65°Step 2) We move 3 units along the z-axis, as shown in the diagram below.

Step 3) We rotate 40° from the reference axis (x) in a counterclockwise direction, as shown in the diagram below.

By convention, the counterclockwise direction is considered positive, and the clockwise direction is negative.

Step 4) Now, we rotate 65° from the z-axis. The point where the line ends is the point with spherical coordinates (3, 40°, 65°). We plot the point there, as shown in the diagram below.The spherical coordinate system is an extension of the two-dimensional polar coordinate system. It can also be extended to higher-dimensional spaces and is then referred to as a hyperspherical coordinate system.

## Conversion Between Spherical and Cartesian Coordinates

### 1) How to convert from spherical to cartesian coordinates

We can convert from spherical to cartesian coordinates using the following equations:

x = ρ sin(φ) cos(θ)

y = ρ sin(φ) sin(θ)

z = ρ cos(φ)

### Converting Spherical to Cartesian Coordinates Example 2.1

Convert the spherical coordinates (5, 50°, 35°) to cartesian coordinates.

Step 1) We identify the ρ, θ, and φ coordinates as:

ρ-coordinate = 5 θ-coordinate = 50° φ-coordinate = 35°Step 2) We use the previous equations to find the x, y, and z coordinates:

x = ρ sin(φ) cos(θ) = 5 sin(35°) cos(50°) ≈ 1.8 y = ρ sin(φ) sin(θ) = 5 sin(35°) sin(50°) ≈ 2.2 z = ρ cos(φ) = 5 cos(35°) ≈ 4.1The point with spherical coordinates (5, 50°, 35°) has cartesian coordinates (1.8, 2.2, 4.1), as shown in the diagram below.

### 2) How to convert from cartesian to spherical coordinates

The cartesian coordinates x, y, and z can be converted to spherical coordinates ρ, θ, and φ with ρ ≥ 0 and θ in the interval (0, 2π) by:

π is equal to 180°.

### Converting Cartesian to Spherical Coordinates Example 2.2

Convert the cartesian coordinates (-2, 1, 3) to spherical coordinates (ρ, θ, φ).

Step 1) We identify the x, y, and z coordinates as:

x-coordinate = -2 y-coordinate = 1 z-coordinate = 3Step 2) Since the x-coordinate is negative and the y-coordinate is positive, the angular coordinate θ can be found as:

θ = arctan(y/x) + 180° = arctan(1/-2) + 180° = 153.4°Step 3) We use the Pythagorean theorem to find the radial distance (ρ) coordinate as:

Step 3) Now, we can find the angular coordinate φ using the following equation:

The point with cartesian coordinates (-2, 1, 3) has spherical coordinates (3.7, 153.4°, 35.8°), as shown in the diagram below.

## Conversion Between Spherical and Cylindrical Coordinates

### 1) How to convert from spherical to Cylindrical coordinates

We can convert from spherical to Cylindrical coordinates using the following equations:

r = ρ sin(φ)

θ = θ

z = ρ cos(φ)

### Converting Spherical to Cylindrical Coordinates Example 3.1

Convert the spherical coordinates (8, 50°, 60°) to Cylindrical coordinates (x,y,z).

Step 1) We identify the ρ, θ, and φ coordinates as:

ρ-coordinate = 8 θ-coordinate = 50° φ-coordinate = 60°Step 2) We use the previous equations to find the r, θ, and z coordinates:

r = ρ sin(φ) = 8 sin(60°) ≈ 6.9 θ = 50° z = ρ cos(φ) = 8 cos(60°) = 4The point with spherical coordinates (8, 50°, 60°) has cylindrical coordinates (6.9, 50°, 4), as shown in the diagram below.

### 2) How to convert from cylindrical to spherical coordinates

The cylindrical coordinates r, θ, and z can be converted to spherical coordinates ρ, θ, and φ using the following equations:

θ = θ

### Converting Cylindrical to Spherical Coordinates Example 3.2

Convert the cylindrical coordinates (4, 45°, 3) to spherical coordinates (ρ, θ, φ).

Step 1) We identify the r, θ, and z coordinates as:

r-coordinate = 4 θ-coordinate = 45° z-coordinate = 3Step 2) We use the previous equations to find the ρ, θ, and φ coordinates:

θ = 45°

The point with cylindrical coordinates (4, 45°, 3) has spherical coordinates (5, 45°, 53.1°), as shown in the diagram below.

## Equations in Spherical Coordinates

Spherical coordinates provide a more simple way for describing and analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet's atmosphere. For example, a sphere that has the Cartesian equation x^{2}+y^{2}+z^{2}=c^{2} has the simple equation ρ=c in spherical coordinates.

### Describing Surfaces Using Spherical Coordinates Example 4.1

Try to describe the surfaces with the following spherical equations:

a) θ = 30° b) φ = 45° c) ρ = 5

a) When the angle θ is held constant while ρ and φ are allowed to vary, the result is a half-plane (see the diagram below).

b) Equation φ = 45° describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring 45° with the positive z-axis. These points form a half-cone (see the diagram below).

b) The equation ρ = 5 describes the set of all points 5 units away from the origin, or in other words, a sphere with radius 5 (see the diagram below).

### Frequently Asked Questions

- What are spherical coordinates?
- Spherical coordinates are a generalization of the two-dimensional polar coordinates that allow us to describe the position of points in 3d space.

- When to use spherical coordinates?
- Spherical coordinates are useful in describing and analyzing systems that have some degree of symmetry about a point, such as spheres, the potential energy field surrounding a concentrated mass or charge, or global weather simulation in a planet's atmosphere.

**References:**Gilbert Strang. “Calculus Volume 3.” OpenStax, 2016. March 30. https://openstax.org/books/calculus-volume-3/pages/2-7-cylindrical-and-spherical-coordinates.