--- title: "Coons patch definition and point calculation" subject: "Computer Aided Design" chapter: 4 key: "semester_4/computer_aided_design:coons-patch-numerical" source: https://www.bctengineer.com/q/semester_4/computer_aided_design/coons-patch-numerical site: BCT Engineering --- # Coons patch definition and point calculation Computer Aided Design · Chapter 4: Representation of Surfaces and Solids - Asked in: year not given, 8 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/coons-patch-numerical - Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-4 ## Question Explain the Coons patch and the bilinearly blended Coons surface formula. A patch has the boundary curves P(u,0) = (4u, 0, 0), P(u,1) = (4u, 3, 2u^2), P(0,w) = (0, 3w, 0) and P(1,w) = (4, 3w, 2w^2). Check the corner compatibility and calculate the surface point at (u, w) = (0.5, 0.5) and (0.5, 0.25). ## Short answer A Coons patch is defined by four boundary curves and bilinearly blended by P=ruled boundary terms minus corner bilinear term. Corners agree at (0,0,0), (4,0,0), (0,3,0), (4,3,2), so the patch is valid. The required points are P(0.5,0.5)=(2,1.5,0) and P(0.5,0.25)=(2,0.75,-0.0625). ## Answer 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).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What defines a Coons patch? A. A net of control points that the surface only approximates B. Its four boundary curves, which the patch passes through exactly C. Two knot vectors D. A profile swept along a path Answer: B. A Coons patch blends four boundary curves, so the edges are reproduced exactly. (Key point: Coons patch idea) 2. Why is the bilinear corner term subtracted in the bilinearly blended Coons formula? A. To make the surface smaller B. Because the two ruled surfaces together count the corner contributions twice C. To remove the z-component D. To make the boundary curves straight Answer: B. The sum of the two ruled surfaces double counts the bilinear surface through the corners, so it is subtracted. (Key point: Corner term in formula) 3. Ruled terms at (0.5, 0.5) sum to (4, 3, 0.5) and the corner term is (2, 1.5, 0.5). What is P(0.5, 0.5)? A. (6, 4.5, 1) B. (2, 1.5, 0) C. (2, 1.5, 0.5) D. (4, 3, 0) Answer: B. Subtract the corner term from the ruled terms: (4-2, 3-1.5, 0.5-0.5) = (2, 1.5, 0). (Key point: Result at (0.5, 0.5)) 4. At (0.5, 0.25) the ruled terms are (4, 1.5, 0.1875) and the corner term is (2, 0.75, 0.25). What is the z of the surface point? A. 0.4375 B. 0.25 C. 0 D. -0.0625 Answer: D. z = 0.1875 - 0.25 = -0.0625, so P = (2, 0.75, -0.0625). (Key point: Result at (0.5, 0.25)) ## Related in Computer Aided Design - [Biquadratic Bezier surface point calculation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-surface-numerical) — Computer Aided Design, chapter 4 - [Cubic Bezier point and tangent calculation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-curve-numerical) — Computer Aided Design, chapter 3 - [Ruled surface between two lines: points](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Aruled-surface-numerical) — Computer Aided Design, chapter 4 - [Geometry and topology; Euler check of solid](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Ageometry-topology-euler) — Computer Aided Design, chapter 2 ## Related in other subjects - [Laplace: five-point formula, 2x2](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-five-point-iterative) — Numerical Method, chapter 7 - [Point to point method of lighting calculation](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_power_distribution_and_illumination%3Apoint-to-point-method) — Industrial Power Distribution and Illumination, chapter 7 - [Boundary layer separation and stagnation point](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Abl-separation-stagnation-point) — Fluid Mechanics, chapter 8 - [Resultant force on curved surface ABCDE (S = 1.25)](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Acurved-surface-abcde-s125-10m) — Fluid Mechanics, chapter 3