Coons patch definition and point calculation

A Coons patch is a surface patch defined by its four boundary curves (not by a net of control points). The interior is obtained by blending the boundaries, so the patch exactly passes through all four edges. It is used to fill a four-sided region bounded by given curves, such as a gap between existing surfaces.

Bilinearly blended Coons surface

$$\begin{aligned} \mathbf P(u,w)={}&(1-w)\mathbf P(u,0)+w\,\mathbf P(u,1)+(1-u)\mathbf P(0,w)+u\,\mathbf P(1,w) \\ &-\big[(1-u)(1-w)\mathbf P_{00}+u(1-w)\mathbf P_{10}+(1-u)w\,\mathbf P_{01}+uw\,\mathbf P_{11}\big] \end{aligned}$$

The first four terms are two ruled surfaces; the bracket (the bilinear surface through the corners) is subtracted because the corners are counted twice.

Corner compatibility

$\mathbf P_{00}=(0,0,0)$, $\mathbf P_{10}=(4,0,0)$, $\mathbf P_{01}=(0,3,0)$, $\mathbf P_{11}=(4,3,2)$.

The curves agree at every corner (e.g. $P(u{=}1,w{=}1)=(4,3,2)$ from both the top and right curves), so the patch is valid.

At (0.5, 0.5)

Boundary points: $\mathbf P(0.5,0)=(2,0,0)$, $\mathbf P(0.5,1)=(2,3,0.5)$, $\mathbf P(0,0.5)=(0,1.5,0)$, $\mathbf P(1,0.5)=(4,1.5,0.5)$.

Ruled terms: $0.5(2,0,0)+0.5(2,3,0.5)+0.5(0,1.5,0)+0.5(4,1.5,0.5)=(4,3,0.5)$

Corner term: $0.25[(0,0,0)+(4,0,0)+(0,3,0)+(4,3,2)]=0.25(8,6,2)=(2,1.5,0.5)$

$$\mathbf P(0.5,0.5)=(4,3,0.5)-(2,1.5,0.5)=(2,\ 1.5,\ 0)$$

At (0.5, 0.25)

Boundary points: $\mathbf P(0.5,0)=(2,0,0)$, $\mathbf P(0.5,1)=(2,3,0.5)$, $\mathbf P(0,0.25)=(0,0.75,0)$, $\mathbf P(1,0.25)=(4,0.75,0.125)$.

Ruled terms: $0.75(2,0,0)+0.25(2,3,0.5)+0.5(0,0.75,0)+0.5(4,0.75,0.125)=(1.5+0.5+0+2,\ 0.75+0.375+0.375,\ 0.125+0.0625)=(4,\ 1.5,\ 0.1875)$

Corner term: $(0.5)(0.75)(0,0,0)+(0.5)(0.75)(4,0,0)+(0.5)(0.25)(0,3,0)+(0.5)(0.25)(4,3,2)=(1.5+0.5,\ 0.375+0.375,\ 0.25)=(2,\ 0.75,\ 0.25)$

$$\mathbf P(0.5,0.25)=(4,1.5,0.1875)-(2,0.75,0.25)=(2,\ 0.75,\ -0.0625)$$

Answer: P(0.5, 0.5) = (2, 1.5, 0); P(0.5, 0.25) = (2, 0.75, -0.0625).