subject: Spherical Coordinates Derivation [print this page] In mathematics and its applications, the choice of co-ordinate system is very important in determining how tedious your work is going to be. Take the example for equation of a circle.
In Cartesian system of co-ordinates, its written as
x2 + y2 = a2 (center is at origin and radius is 'a')
while in polar coordinates, it can simply be written as
r = a.
In three dimensions, the choice of co-ordinates becomes even more important. Therefore it makes it easier if we are acquainted with various systems. Here we will discuss Spherical co-ordinate system.
In spherical system of co-ordinates, the point is represented by its distance from origin, and two angles, called the azimuth and inclination.
Derivation of Spherical Coordinates System
To derive the relationship between Cartesian and spherical coordinates lets look closely at this picture.
spherical coordinates
Now in this picture, the coordinates x, y and z of yellow dot are represented by lines rx, ry and rz. Now the angle T is called the inclination and f is called the azimuth. From this figure, rz is simply projection of r on the z axis and is written as
rz = r cosT.
The dotted line on xy plane is projection of r on the xy plane. Lets name it rxy and its length is
rxy = r sinT.
The angle between x axis and rxy is f. The projection of rxy on x axis is rx which can be written as
rx = rxy cos f = r sinT cosf.
Similarly,
ry = rxy sinf = r sinT sinf.
So we see that we can write the spherical coordinates as
rx = r sinT cosf.
ry = r sinT sinf.
rz = r cosT.
Solved Examples
1) Lets apply this formula to a point (20, 30o, 45o) in spherical and find its (X, Y, Z) co-ordinates.
If the length of all the three sides of the triangle is equal, then it is called as an equilateral triangle. The perimeter of an equilateral triangle is 3a (where a is the length of the side). Its area is given by = sqrt(3/4) xx a^2.
Now let us see few problems on this topic equilateral triangle co-ordinates. To find the length we use distance formula sqrt[ (x 2 ** x 1)^2 + (y2 ** y1)^2] .
Example Problems on Equilateral Triangle Coordinates
Ex 1. Show that the co-ordinates A (3, 2), B(7, 4) and C (5 + sqrt 3, 3 -2 sqrt 3) are the vertices of an equilateral triangle. Then find its perimeter and the area.
Soln: Given: A (3, 2), B (7, 4), C (5 + sqrt 3 , 3 2 sqrt 3 )
