Spherical coordinate system

In mathematics, a spherical coordinate system is a way to describe the position of a point in three-dimensional space using three numbers. This system is very useful for problems that have spherical symmetry, like studying planets or sound waves.

The system uses three values: the radial distance r, which tells how far the point is from a fixed starting point called the origin; the polar angle θ, which tells how far the point is tilted from a special axis; and the azimuthal angle φ, which tells how much the point has rotated around that axis. These three values together, called a 3-tuple, can pinpoint any point on a sphere.

This coordinate system is often used in physics and engineering because it simplifies calculations for objects that are round or symmetrical. For example, it helps scientists understand how light or sound spreads out from a source in all directions. The plane that passes through the origin and is perpendicular to the polar axis is known as the reference plane.

Terminology

In this article, we follow the physics convention for spherical coordinates. The distance from the origin to a point is called the radius or radial distance. The angle between this line and a fixed axis is called the polar angle, but it can also be named inclination, zenith, normal, or colatitude. Sometimes people use the elevation angle instead, which measures upward from the reference plane.

Different fields use different symbols and orders for these coordinates. In physics, the order is usually radial distance, polar angle, and azimuthal angle, written as (r, θ, φ). In mathematics, the order can be different, switching the meanings of the angles. It’s important to check which convention a source is using. Geographical systems use latitude, longitude, and height, while celestial systems have their own terms. These systems often use radians instead of degrees.

Definition

To define a spherical coordinate system, you need to pick a starting point called the origin and two directions: one straight up (zenith) and one to the side (azimuth). These help create a flat reference plane, usually thought of as horizontal.

The spherical coordinates of any point are:

• The radius, which is how far the point is from the origin.
• The inclination (or polar angle), which tells how far up or down the point is from the zenith direction.
• The azimuth (or azimuthal angle), which shows how much the point turns around from the azimuth reference direction.

Angles can be measured in degrees or radians, and the system can be used to describe points on a sphere or in three-dimensional space.

Coordinate system conversions

See also: List of common coordinate transformations § To spherical coordinates

The spherical coordinate system is one way to describe a point in space, and we can change these descriptions into other systems, like Cartesian or cylindrical coordinates.

For example, if we know a point’s Cartesian coordinates (x, y, z), we can find its spherical coordinates (r, θ, φ) using special math rules. Similarly, cylindrical coordinates (ρ, φ, z) can also be changed into spherical coordinates, and vice versa. These changes help scientists and engineers solve problems in many fields.

Ellipsoidal coordinates

See also: Ellipsoidal coordinates

You can also use a special version of spherical coordinates to describe points on an ellipsoid, which is like a stretched sphere.

The coordinates tell you how far away the point is and the angles from special lines, but they are changed a little bit to fit the shape of the ellipsoid. This helps scientists and engineers work with these shapes more easily.

Integration and differentiation in spherical coordinates

The spherical coordinate system uses three values to describe a point in space: the distance from the origin (r), the angle from the positive z-axis (θ), and the angle around the z-axis (φ). These coordinates are useful for problems that have spherical symmetry.

Working with these coordinates involves special formulas for calculating small changes and integrals. For example, the smallest piece of space (an "infinitesimal volume") in spherical coordinates has a specific size that depends on all three coordinates. This makes it possible to add up tiny pieces to find the total for larger spaces or to describe how quantities change in spherical patterns.

Distance and angle in spherical coordinates

In spherical coordinates, we can find the distance and angle between two points using their positions. The distance depends on their radial distances and the angles between their positions.

The angle between the two points can also be calculated using their coordinates, which involves some trigonometry. This helps us understand how points are arranged in three-dimensional space using distances and angles.

Main article: Angle difference identity

Kinematics

In spherical coordinates, we describe the position, movement, and forces of objects in three-dimensional space using three values: distance from a central point and two angles.

The velocity of an object in these coordinates depends on how quickly these distance and angles change over time. Similarly, its acceleration depends on the rates of change of its velocity. These ideas help scientists and engineers understand motion in three dimensions, like the path of a planet or a moving car.