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 a three dimensional space, a point is uniquely defined 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 which can be obtained by drawing three lines in space. All three points cross a point that is commonly termed as the origin and are 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 are 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 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\Theta, φ), where r is radial coordinate, Θ\Theta is polar angle and φ is known as 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\Theta ≤ 180° and 0° ≤ φ < 360°.

In short, In spherical coordinate system, each component of the vector is the function of θ\theta, ϕ\phi and r.

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

x=rsinθcosϕx = r \sin \theta \cos \phi,

y=rsinθsinϕy = r \sin \theta \sin \phi,

z=rcosθz = r \cos \theta.

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

Rectangular to Spherical

r2 = x2 + y2 + z2,

tan Φ = y/x and

Θ\Theta= =cos1(z/x2+y2+z2=cos^{-1}(z /\sqrt{x^2 + y^2 + z^2}

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 A=A1r+A2θ+A3ϕA = A_1r + A_2 \theta + A_3 \phi is in spherical coordinate system.

In spherical coordinate system, unit vectors are r^\hat{r},θ^\hat{\theta },ϕ^\hat{\phi } and in terms of rectangular system is as below,

r^=x^sinθcosϕ+y^sinθsinϕ+z^cosθ\hat{r} = \hat{x} \sin \theta \cos \phi + \hat{y} \sin \theta \sin \phi + \hat{z} \cos \theta θ^=x^cosθcosϕ+y^cosθsinϕz^sinθ\hat{\theta} = \hat{x} \cos \theta \cos \phi + \hat{y} \cos \theta \sin \phi – \hat{z} \sin \theta ϕ^=x^sinϕ+y^cosϕ\hat{\phi} = -\hat{x} \sin \phi + \hat{y} \cos \phi

Also read

Types of coordinate systems

Cartesian coordinate system

Solved Example

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

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

r2 sin2 Θ\Theta cos2Φ + r2 sin2 Θ\Theta sin2 Φ = r2 cos2 Θ\Theta

r2 sin2 Θ\Theta (cos2Φ + sin2 Φ) = r2 cos2 Θ\Theta

r2 sin2 Θ\Theta = r2 cos2 Θ\Theta (Here r ≥ 0)

tan2 Θ\Theta = 1

So, Θ\Theta = π/4 or Θ\Theta = 3π/4

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