Converting Cartesian Coordinates

We can convert Cartesian coordinates systems to polar and spherical coordinates using the derived formulae.  .

In honor of Descartes, the system used to depict the location of a point in a plane is also called as the Cartesian coordinates system.

In polar coordinates systems, the distance is used to determine the points from a fixed point and the direction is used to determine the angle.

In spherical coordinates systems the position of a point is shown by three  numbers.

Method on converting cartesian coordinates:

Converting Cartesian coordinates  to polar:

Cartesian coordinates are denoted as (x, y) and polar co ordinates are as (r, θ) .  The following formula is used for converting cartesain coordinates to polar,

r = sqrt(x^2+y^2)

Where,

r = distance from origin to the point

x = Cartesian x-coordinate

y = Cartesian y-coordinate

We can find θ using tangent function,

θ = tan -1 (y/x)    {tan θ = y/ x}

Where,Θ= angle relative to the zero axis.

Converting Cartesian Coordinates  to spherical:

Spherical coordinates are denoted by (ρ, Φ, θ). The formulae used for converting cartesain coordinates to Spherical are,

To find ρ,Φ,θ::

ρ  =  sqrt(x^2+y^2+z^2)

Φ =  cos-1 (z/ ρ)

θ =  y/ ρ sin Φ .

Where , ρ = distance , Φ = angle need to rotate , θ = value of angle need to rotate .


Example problems on converting cartesian coordinates:

Example for converting Cartesian coordinates  to polar:

Find the polar form of   (15, 8).

Solution:Step 1: Find distance r

r  = sqrt(x^2+y^2)

= sqrt(15^2+8^2)

= sqrt(225+64)

= sqrt(289)        = 17

Step 2: Find angle

θ= tan -1 (y/x)

= tan -1 ( 8/15)

= tan -1 (0.533)

= 28.05 ◦

The polar form is (17, 28.05◦) {Rounded to the nearest integer}

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Example for converting Cartesian coordinates  to spherical:

Convert the points (8, 4, 2) from Cartesian to spherical form?

Solution:X=8, Y=4, Z=2

To find ρ:ρ      = sqrt(x^2+y^2+z^2)

=sqrt(8^2+4^2+2^2)

=sqrt(84)

=9.165

To find Φ:Φ = cos-1 (z/ ρ)

= cos-1 (2/ 9.165)

= cos-1 (0.2182)

= 77. 39°

To find θ:θ = y/ ρ sin Φ .

=  4/ (9.165 sin (77.39))

=  4/ (9.165 x 0.9758)

=  4/ 8.9439

= 0.4472°

Spherical coordinates is( 9.165, 77.39°, 0.4472°)