Explanation:
Given;
- The equation of the surface is, ρ = sin(θ) sin(φ)
The spherical and Cartesian coordinates are related by the equations,
x = ρ sin(φ) cos(θ)
y = ρ sin(φ) sin(θ)
z = ρ cos(φ)
Here, x, y and z represent the coordinates of the center of the sphere on x, y and z axes respectively.
Now, multiply the given surface equation by ρ.
ρ² = ρsin(θ) sin(φ)
We have, ρ² = x² + y² + z².
Substitute this value in above equation and also replace the RHS by y coordinate.
x² + y² + z² = y
Now, simplify the above equation.
x² + y² - y + z² = 0
x² + y² - 2y½ + (1/2)² + z² = (1/2)²
x² + (y - 1/2)² + z² = (1/2)² ...(1)
The general form of equation of sphere is,
(x - a)² + (y - b)² + (z - c)² = r²
Here, a, b, c are the coordinates of the center of the sphere on x, y, z axes respectively and r is the radius of the sphere.
Compare equation (1) with the general form of equation of sphere. Therefore we get,
Therefore, the radius of the sphere is 1/2 and center of sphere is at (0, 1/2, 0).
Thus, The given equation represents - a sphere of radius 1/2 centers at (0, 1/2, 0).