Spatial Bi-linear Interpolation

Q(a) = P₁ + (P₂ - P₁)a(1)

R(a)=P₄ + (P₃ - P₄)a(2)

P(a,b) = Q + (R - Q)b(3)

P(a,b) = P₁ + (P₂ - P₁)a + [(P₄ + (P₃ - P₄)a) - (P₁ + (P₂ - P₁)a)]b(4)

P(a,b)=P₁ + (P₂ - P₁)a + (P₄ - P₁)b + (P₁ - P₂ + P₃ - P₄)ab(5)

P(x) = x₁ + (x₂ - x₁)a + (x₄ - x₁)b + (x₁ - x₂ + x₃ - x₄)ab(6)

P(y) = y₁ + (y₂ - y₁)a + (y₄ - y₁)b + (y₁ - y₂ + y₃ - y₄)ab(7)