Relation Between Rectangular and Spherical Coordinate System

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The well-defined set of values that explain the location of any given point in space refers to coordinates. Two coordinates represent any arbitrary point in a two-dimensional system. Similarly, in a three-dimensional space, a point is uniquely represented by three coordinates. There are different ways of describing the location of a point in 2-D or 3-D space like cartesian, spherical, and more. In this handout, you will get to know more about rectangular or cartesian coordinate, spherical coordinate, and the relation between rectangular and spherical coordinate.


Rectangular Coordinate System- What is it?

The cartesian or rectangular coordinate system provides a straightforward way to define the location of points in three-dimensional space. All the three lines passing across a point are perpendicular to each other and cross a point, commonly known as the origin. These three lines are called axes. Let a point is represented by (x, y, z) in a 3-D system, and three planes XY, YZ, and ZX define each pair of axes. 

(Image to be added soon)

The above picture shows a point in three-dimensional space that is represented by three coordinates, namely- x, y, and z. These three coordinates (x, y, z) form a rectangular or cartesian coordinate system. 


Spherical Coordinate System- What is it?

A coordinate system with a static origin plus a zenith direction refers to a spherical coordinate system. Zenith is an imaginary point that is present directly above the origin. The direction of the zenith point refers to the direction from the origin to the zenith. Every arbitrary point in 3-D space has three spherical coordinates namely- r as radius, Ɵ as the polar angle, and ø as the azimuthal angle.

  • The radius (r) is the distance between the arbitrary point to the origin. 

  • The polar angle (Ɵ) is the angle made by the line connecting the arbitrary point with the origin in the zenith direction. 

  • The azimuthal angle (ø) is the angle between the perpendicular projection of the point to the plane and a fixed reference direction on an arbitrary point. 

(Image to be added soon)

The above diagram shows a point in three-dimensional space represented by spherical coordinates. The diagram represents a relation between rectangular and spherical coordinates. 

So, any coordinate system for 3-D space where the position of any arbitrary point is specified by spherical coordinates (r, Ɵ, ø) refers to a spherical coordinate system. To define a unique set of (r, Ɵ, ø) for each point, the constraint of ranges:

r ≥ 0;  0o ≤ Ɵ ≤ 180o;  0o ≤ ø < 3600


The Relation Between Rectangular and Spherical Coordinate

To define the relation between rectangular and spherical coordinate, the formula used:

r = \[\sqrt{x^{2}+y^{2}+z^{2}}^{2}\]

So, rectangular coordinates will be given as:

x = r sinƟ cosø

y = r sinƟ sinø

z = r cosø

From the relation between rectangular and spherical coordinate, spherical coordinate can be expressed in terms of x, y, and z as:

r = \[\sqrt{x^{2}+y^{2}+z^{2}}\]

tan ø = y/ x

ø = tan-1 (y/ x)

cosƟ = (z / \[\sqrt{x^{2}+y^{2}+z^{2}}\] )

Ɵ = cos-1 (z / \[\sqrt{x^{2}+y^{2}+z^{2}}\] )


How to Perform Dot Product in the Spherical Coordinate System?

The dot product of the given vector and a unit vector in the desired direction is taken to find the desired component of a vector in a spherical coordinate system. The relation between rectangular and spherical coordinate shows if a vector E = E1x + E2y + E3z is in a rectangular coordinate system then E = E1r + E2Ɵ + E3ø is in a spherical coordinate system.

Now, in a spherical coordinate system, unit vectors are defined as r^+ Ɵ^+ ø^

So, from the relation between rectangular and spherical coordinate, the given vectors r^+ Ɵ^+ ø^ can be represented in the rectangular system as:

r^ = x^ sinƟ cosø + y^ sinƟ sinø + z cosƟ

Ɵ^ =x^ cosƟ cosϕ + y^​cosƟ sinϕ –z^ sinƟ

Ø = -x^ sinø + y^ cosø


Take A Look At One of The Solved Examples:

Question:  Use the relation between rectangular and spherical coordinates to convert (2, 4, 5) in rectangular coordinates to spherical coordinates.

Solution: From the relation between rectangular and spherical coordinate, we know that

r = \[\sqrt{x^{2}+y^{2}+z^{2}}\]

ø = tan-1 (y/ x)

Ɵ = cos-1 (z / \[\sqrt{x^{2}+y^{2}+z^{2}}\] )

From the question, x = 2, y = 4, and z = 5; 

Put the values of x, y, and z in the above three equations:

r = \[\sqrt{2^{2}+4^{2}+5^{2}}\] 

r = \[\sqrt{45}\]

ø = tan-1 (4/ 2) = tan-1 (2)

Ɵ = cos-1 (5 / \[\sqrt{45}\] )

FAQ (Frequently Asked Questions)

1. What is the Meaning of Phi, Rho, and Theta in the Spherical Coordinates?

In the spherical coordinates, Rho will define the distance from the starting point. However, theta represents the angle to define between polar coordinates. Phi will define the angle made by the Z-axis with the line that connects the point with the origin. Thus it helps in defining the relation between rectangular and the spherical coordinate system. These three are the main components to define the relations of a coordinate system with a specified angle. Phi is represented in the X-Y plane to define sine, cos, and tan.

2. How Do You Know Usage for Spherical and Cylindrical Coordinates?

If you are trying to solve for the spherical and the cylindrical surfaces then better you go with spherical and the cylindrical coordinates. These coordinates work efficiently on the region of integration rather than using the function. Thus it gives better visualisation of the diagram, and you efficiently define the coordinate system. Using cylindrical coordinates could be very flexible. Spherical coordinates create a mess at last. In case you feel everything to be messy, better go for cylindrical coordinates. Or you can also use spherical coordinates at the first and cylindrical at the second.