Spherical coordinate system

In mathematics, a spherical coordinate system is a way to show where a point is in three-dimensional space using three numbers. This system works well for round objects, like planets or sound waves.

The system uses three values: the radial distance r, which shows how far the point is from a starting point called the origin; the polar angle θ, which shows how much the point is tilted from a special line; and the azimuthal angle φ, which shows how much the point has turned around that line. These three values together can find any point on a sphere.

This coordinate system is often used in physics and engineering. It makes it easier to study objects that are round or symmetrical. For example, it helps scientists learn how light or sound moves out from a source in all directions. The flat area that goes through the origin and is perpendicular to the polar axis is called the reference plane.

Terminology

In this article, we use the way physicists talk about spherical coordinates. The distance from the starting point to another point is called the radius or radial distance. The angle between this line and a fixed axis is called the polar angle. It can also be called inclination, zenith, normal, or colatitude. Some people use the elevation angle instead, which measures how high something is from a flat surface.

Different areas 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 math, the order can be different, changing what the angles mean. It’s important to see which way a source is using. Geography uses latitude, longitude, and height, while space systems have their own names. These systems often use radians instead of degrees.

Definition

To use a spherical coordinate system, you need a starting point called the origin and two directions: one straight up (zenith) and one to the side (azimuth). These help make 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. 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 use a special kind of spherical coordinates to talk about points on an ellipsoid, which is a sphere that has been stretched.

The coordinates tell you how far away a point is and the angles from special lines. They are changed a little to match 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 a round shape.

Working with these coordinates uses special formulas. For example, the smallest piece of space in spherical coordinates has a size that depends on all three coordinates. This helps us add up tiny pieces to find the total for larger spaces or to describe how things change in round patterns.

Distance and angle in spherical coordinates

In spherical coordinates, we can find the distance and angle between two points by looking at where they are placed. The distance changes based on how far out each point is and the angles between them.

We can also work out the angle between two points using their coordinates and some basic math. This shows us how points are set out in three-dimensional space using just distances and angles.

Main article: Angle difference identity

Kinematics

In spherical coordinates, we describe where an object is and how it moves in three-dimensional space using three values: distance from a central point and two angles.

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