Coons patch

Given four space curves c₀(s), c₁(s), d₀(t), d₁(t) which meet at four corners c₀(0) = d₀(0), c₀(1) = d₁(0), c₁(0) = d₀(1), c₁(1) = d₁(1); linear interpolation can be used to interpolate between c₀ and c₁, that is

${\displaystyle L_{c}(s,t)=(1-t)c_{0}(s)+tc_{1}(s)}$

and between d₀, d₁

${\displaystyle L_{d}(s,t)=(1-s)d_{0}(t)+sd_{1}(t)}$

producing two ruled surfaces defined on the unit square.

The bilinear interpolation on the four corner points is another surface

${\displaystyle B(s,t)=c_{0}(0)(1-s)(1-t)+c_{0}(1)s(1-t)+c_{1}(0)(1-s)t+c_{1}(1)st.}$

A bilinearly blended Coons patch is the surface

${\displaystyle C(s,t)=L_{c}(s,t)+L_{d}(s,t)-B(s,t).}$

Although the bilinear Coons patch exactly meets its four boundary curves, it does not necessarily have the same tangent plane at those curves as the surfaces to be joined, leading to creases in the joined surface along those curves. To fix this problem, the linear interpolation can be replaced with cubic Hermite splines with the weights chosen to match the partial derivatives at the corners. This forms a bicubically blended Coons patch.

• Surface
• Atlas (topology)
• Interpolation

• Weiqing Gu. "Surface Construction Schemes". Retrieved 8 April 2012.