section_path: Construction of simple lenses/For a spherical surface/Derivation
section_name: Derivation
paragraph_text: Derivation
  
  Applying Snell's law on the spherical surface, n 1 sin i = n 2 sin r. {\displaystyle n_{1}\sin i=n_{2}\sin r\,.}
  
  Also in the diagram, tan (i − θ) = h u tan (θ − r) = h v sin θ = h R {\displaystyle {\begin{aligned}\tan(i-\theta)&={\frac {h}{u}}\\\tan(\theta -r)&={\frac {h}{v}}\\\sin \theta &={\frac {h}{R}}\end{aligned}}}, and using small angle approximation (paraxial approximation) and eliminating i, r, and θ,
  
  n 2 v + n 1 u = n 2 − n 1 R. {\displaystyle {\frac {n_{2}}{v}}+{\frac {n_{1}}{u}}={\frac {n_{2}-n_{1}}{R}}\,.}