The space itself is 3-dimensional.  That means it has 3 degrees of freedom.
Any point in the space is determined by three free variables x, y and z.

Any equation between these 3 variables, puts a constraint on the points of the space,
and deducts 1 degree of their freedom. The result is a 2-dimensional subspace.

Surface
any 2-dimensional subspace of "the space" is called a surface.
That means any point on the surface is determined by 2 parameters.




Sphere
with center at O(0,0,0) and of radius r

Assume A(x,y,z) be an arbitrary point on its surface
and H(x,y,0) be the vertical image 0f A on xy plane

θ=∠xOH is the longitude of A
φ=∠HOA is it latitude


  
    
      
    
    
  

  
  
  

  
  z

  
  y

  
  x

  
  O

  
  A

  
  r

  
  H

  
  

  
  z

  
  y

  
  x

  
  θ

  
  φ

  


  
    z = r sin φ
    OH = r cos φ
  

  
    x = OH cos θ
    y = OH sin θ
This is the parametric equation of the sphere
\( \begin{cases} x(\theta, \phi) = r \cos\theta \cos\phi & (-\pi \lt \theta \le \pi) \\ y(\theta, \phi) = r \sin\theta \cos\phi & (-\frac{\pi}{2} \le \phi \le \frac{\pi}{2}) \\ z(\theta, \phi) = r \sin\phi \end{cases} \)

by eliminating \( \theta \) and \(\phi\) between \(x, y\) and \(z\),
we reach to the implicite equation of the sphere:
\( x^2 + y^2 + z^2 = r^2 \)





1731 Clairaut: curves and surfaces in 3-dimensional space
1748 Euler: 2nd degree surfaces