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°).

Solution

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.

Solution

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°) ≈ 4.1

The 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).

Solution

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

x-coordinate = -2 y-coordinate = 1 z-coordinate = 3

Step 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).

Solution

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°) = 4

The 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).

Solution

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

r-coordinate = 4 θ-coordinate = 45° z-coordinate = 3

Step 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 x²+y²+z²=c² 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

Solution

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.