User:Frank Ford Little Ph.D.

Coons' Patch

Steven Anson Coons' Patch creates a surface from curves in Geometric Design. Its trivial formula belies its seminal importance. Many airplanes, automobiles, appliances and parts are Coons' patches. Lofting projectors, polynomial interpolation operators in one variable are cast in two variables by holding one variable fixed yet arbitrary. This partial generalization is labeled “transfinite interpolation” by W.H. Gordon . For formal operators P₁ and P₂ their composition provides the intersection of properties like points of interpolation or function precision. Coons' Boolean Sum gives the union. Like inventing or from and, Coons composed the remainder operators (I-P₁) and (I-P₂) and took the remainder giving: P₁ | P₂ = P₁ + P₂ - P₁P₂

For example, let L₁(u,v) be linear interpolation in u. (v is fixed but arbitrary.) Dually define L₂. L₁L₂ ( = L₂L₁ ) is bilinear polynomial interpolation to the corner data. L₁ | L₂ matches data all around. If F is defined on the unit square,

      L1(F;u,v)       = (1-u)F(0,v) + uF(1,v). 
      L2(F;u,v)      = (1-v)F(u,0) + vF(u,1).
      L1L2(F;u,v)    = (1-u)(1-v)F(0,0) + (1-u)vF(0,1)+
                               u (1-v)F(1,0) + uvF(1,1).

P₁ | P₂ = P₂ + P₁(I-P₂) shows Coons' patch to be “reduction to a previous solution.” Commutivity of the operators is necessary and sufficient for union of properties. The boolean sum, P₁|P₂ has at least the interpolation properties of P₁ and the precision of P₂. If P₂F = F then (P₁|P₂)F = F. If LP₁ = L then L(P₁|P₂) = L.

Coons published Cubic Hermite interpolation in this generalization.