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Coons patch

In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary element method, to mesh problem domains into elements.

Version
v1 · 2026-09-28 · History
Domain-specific #
8715
Domain group
Applied Sciences & Engineering
Origin domain
Computer Science & Software Engineering
Subdomains
Computer Aided Geometric Design, Computer Graphics → Computer Science & Software Engineering

Core Idea

Coons patch is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary element method, to mesh problem domains into elements. In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary.

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Sheet on Four Wires

Imagine four bendy wires joined into a loop, and you want to stretch a smooth sheet so it touches all four wires exactly. A Coons patch is a math recipe for making that sheet. Computer artists use it to fill the gap between shapes so they join up.

Surface From Four Edges

A Coons patch is a way to build a curved surface when you already know the four curves around its edges. Think of four curved fence rails making a loop: the recipe blends the rails together to make a surface that fits every rail exactly. People who make 3D computer graphics use it to join other surfaces together, and engineers use it to split a shape into pieces for computer calculations. One catch: the surface meets the edges exactly, but it may not bend the same way as the neighboring surfaces, so you can get a crease at the seam.

Boundary-Blended Surface Patch

A Coons patch is a way to define a surface from four boundary curves that meet at corners. Think of a parameter grid with coordinates s and t running from 0 to 1. First you blend between one pair of opposite curves along s, then between the other pair along t, and add those two surfaces together. That counts the corners twice, so you subtract the simple bilinear surface built only from the four corner points. The result passes exactly through all four boundary curves. It is used in computer graphics to join surfaces and in engineering simulations (finite and boundary element methods) to split a region into mesh elements. The simple version matches the edges but not necessarily the slope of neighboring surfaces, so creases can appear along the seams.

 

A Coons patch is a surface parametrization over a unit square (s,t) that interpolates four given boundary curves. The bilinear Coons patch is C(s,t) = L_c(s,t) + L_d(s,t) - B(s,t): L_c and L_d are ruled (linear) interpolations between each pair of opposite boundary curves, and B is the bilinear interpolation of the four corner points, subtracted to correct for the corners being included twice. By construction C reproduces all four boundary curves exactly. It is used in computer graphics to join adjacent surfaces smoothly and in computational mechanics, especially finite element and boundary element methods, to mesh problem domains into elements. The bilinear version guarantees only positional continuity: its tangent plane along a boundary need not match that of the surface being joined, which produces creases along the seams. The construction dates to Steven Anson Coons in 1967.

Scope of Application

  • Documented setting. In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in.

  • Bilinear blending. Given four space curves c 0 (s), c 1 (s), d 0 (t), d 1 (t) which meet at four corners c 0 (0) = d 0 (0), c 0 (1) = d.

  • Bilinear blending. B(s,t) = c0(0) (1-s)(1-t) + c0(1) s(1-t) + c1(0) (1-s)t + c1(1) s t.

  • A bilinearly blended Coons patch is the surface. C(s,t)=Lc(s,t)+Ld(s,t)-B(s,t).

  • Bicubic blending. 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.

Clarity

A clear use of Coons patch names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary element method, to mesh problem domains.

Manages Complexity

Coons patch compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—c(s,t)=Lc(s,t)+Ld(s,t)-B(s,t).—and the practical consequence—lc(s,t)=(1-t) c0(s)+ t c1(s). This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit.

Abstract Reasoning

  1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary element method, to mesh problem domains into elements.
  3. Check operation and conditions.

Knowledge Transfer

Within the home domain. Knowledge about Coons patch transfers literally when a new case preserves the same carrier type, relation, and recognition test. In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary element method, to mesh problem domains into elements.

Neighborhood in Abstraction Space

Coons patch sits in a crowded region of the domain-specific corpus (40th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Continuum Mechanics & Field Models (42 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08