Relation between Rectangular and Spherical Coordinate Systems

Introduction to Spherical and Rectangular Coordinates

The sets of values that explain the location of a given point in space are called coordinates. In three-dimensional space, a point is defined uniquely by three coordinates. In this article, we learn about the relation between rectangular and spherical coordinate systems.

The Cartesian system has an alternate name, that is, the rectangular system; it can be obtained by drawing three lines in space. All three points cross a point that is commonly termed the origin and is perpendicular to each other. The lines are called axes of the system. The three planes are XY, XZ and YZ planes that define each pair of axes.

The distance between any arbitrary point and the planes is the coordinates of that point.

A coordinate system with a fixed origin and a zenith direction is a spherical coordinate system. An imaginary point that is present directly above the origin is the zenith. Zenith’s direction is the direction from the origin to the zenith.

Every arbitrary point has three spherical coordinates, namely the radius, the polar angle and the azimuth angle. The radius is the distance between the origin and the point in a three-dimensional space. The polar angle is the angle from the zenith direction and the line which connects the point with the origin. The azimuth angle is the angle between the point’s orthogonal projection to the plane and a fixed reference direction on that plane.

What is the spherical coordinate system? In mathematics, a spherical coordinate system is a coordinate system for 3-dimensional space where the position of a point is specified by a spherical coordinate triplet (r, θ, φ), where r is the radial coordinate, θ is the polar angle, and φ is known as the azimuthal angle. We must draw a constraint of ranges to define a unique set of spherical coordinates for each point. A common choice is r ≥ 0, 0° ≤ θ ≤ 180° and 0° ≤ φ < 360°.

In short, in a spherical coordinate system, each component of the vector is the function of θ, φ and r.

What is a rectangular coordinate system? It is a coordinate system that specifies the location of every point uniquely with a specific set of coordinates. In a two-dimensional plane, the system is based on two perpendicular axes, the x-axis and the y-axis and in three-dimensional space, the system is based on three mutually perpendicular axes, the x-axis, the y-axis and the z-axis.

Relation between the Rectangular Coordinate System and Spherical Coordinate System

Spherical to Rectangular

The relation between the rectangular coordinate system and the spherical coordinate system is

\(\begin{array}{l}x = r \sin \theta \cos \phi\end{array} \)

\(\begin{array}{l}y = r \sin \theta \sin \phi\end{array} \)

\(\begin{array}{l}z = r \cos \theta\end{array} \)

Conversely, these equations can be expressed so that r, θ, Φ can be written in terms of x, y and z. Then, any point with coordinates (x, y, z) has corresponding coordinates (r, θ, Φ).

Rectangular to Spherical

r² = x² + y² + z²,

tan Φ = y/x and

\(\begin{array}{l}\theta = cos^{-1}(z /\sqrt{x^2 + y^2 + z^2})\end{array} \)

Dot Product in Spherical Coordinates

To find the desired component of a vector, we have to take the dot product of the vector and a unit vector in the desired direction.

If a vector A = A₁x + A₂y + A₃z is in a rectangle coordinate system, then

\(\begin{array}{l}A = A_1r + A_2 \theta + A_3 \phi\end{array} \)

is in spherical coordinate system.

\(\begin{array}{l}\text{In spherical coordinate system, unit vectors are}\ \hat{r},\hat{\theta },\hat{\phi }\end{array} \)

and in terms of rectangular system is as below,

\(\begin{array}{l}\hat{r} = \hat{x} \sin \theta \cos \phi + \hat{y} \sin \theta \sin \phi + \hat{z} \cos \theta\\\end{array} \)

\(\begin{array}{l}\hat{\theta} = \hat{x} \cos \theta \cos \phi + \hat{y} \cos \theta \sin \phi – \hat{z} \sin \theta\\\end{array} \)

\(\begin{array}{l}\hat{\phi} = -\hat{x} \sin \phi + \hat{y} \cos \phi\end{array} \)

Also Read

Types of Coordinate Systems

Cartesian Coordinate System

Solved Example

Example: Find an equation in spherical coordinates for the cone surface represented by a rectangular equation, x² + y² = z².

Solution: Substituting the values of x, y, and z, we have

r² sin² θ cos²Φ + r² sin² θ sin² Φ = r² cos² θ

r² sin² θ (cos²Φ + sin² Φ) = r² cos² θ

r² sin² θ = r² cos² θ (Here r ≥ 0)

tan² θ = 1

So, θ = π/4 or θ = 3π/4

The equation θ = 3π/4 represents the lower half-cone, and θ= π/4 represents the upper half-cone.

Frequently Asked Questions

What do you mean by a rectangular coordinate system?

A rectangular coordinate system is a coordinate system that denotes the location of every point with a unique set of coordinates. In the 2D plane, this is based on two axes, i.e., the x-axis and y-axis, and in 3D space, it is based on mutually perpendicular axes (i.e., x-axis, y-axis and z-axis).

What do you mean by abscissa and ordinate?

We represent a point in a two-dimensional plane by (x, y). The x coordinate is called the abscissa, and the y coordinate is called the ordinate.

What do you mean by a spherical coordinate system?

A spherical coordinate system is a coordinate system for 3D space, where the position of a point is denoted by (r, θ, φ), where r is the radial coordinate, θ is the polar angle, and φ is the azimuthal angle.