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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 mathematics_logic_statistics 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 element method, to mesh problem domains into elements. Coons patches are named after Steven Anson Coons, and date to 1967. The bilinear interpolation on the four corner points is another surface.

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) s t. 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.

For Coons patch, the abstraction is narrower than the article's general subject matter: a positive case must preserve 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. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

How would you explain it like I'm…

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.

Structural Signature

Sig role-phrases:

  • Defining carrier — 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) s t.
  • Constitutive relation — C(s,t)=L_c(s,t)+L_d(s,t)-B(s,t).
  • Operating condition — 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.
  • Recognition evidence — 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.
  • Admissible variation — 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 1 (0), c 1 (0) = d 0 (1), c 1 (1) = d 1 (1); linear interpolation can be used to interpolate between c 0 and c 1 , that is.
  • Characteristic consequence — L_c(s,t)=(1-t) c_0(s)+ t c_1(s).
  • Failure boundary — L_d(s,t)=(1-s) d_0(t)+ s d_1(t).

What It Is Not

  • Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by 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.
  • Not an over-broad reading. 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.
  • Not an over-broad reading. 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) s t.
  • Not an over-broad reading. C(s,t)=L_c(s,t)+L_d(s,t)-B(s,t).
  • Not automatically Subdivision surface. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application

Coons patch applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

  • 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 finite element method and boundary element method, to mesh problem domains into elements.
  • 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 1 (0), c 1 (0) = d 0 (1), c 1 (1) = d 1 (1); linear interpolation can be used to interpolate between c 0 and c 1 , that is.
  • Bilinear blending. 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) s t.
  • A bilinearly blended Coons patch is the surface. C(s,t)=L_c(s,t)+L_d(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 creases in the joined surface along those curves.
  • Bicubic blending. 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.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Classification or should be marked as analogy.

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 into elements. The strongest recognition evidence in the frozen account is: 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. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification 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. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity

Coons patch compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—c(s,t)=L_c(s,t)+L_d(s,t)-B(s,t).—and the practical consequence—l_c(s,t)=(1-t) c_0(s)+ t c_1(s). This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning

  1. Type the carrier. Identify the mathematics_logic_statistics 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. 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.
  4. Demand recognition evidence. 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.
  5. Test variation. Change an implementation or setting while preserving 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 1 (0), c 1 (0) = d 0 (1), c 1 (1) = d 1 (1); linear interpolation can be used to interpolate between c 0 and c 1 , that is.
  6. Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
  7. Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Classification.

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. 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 1 (0), c 1 (0) = d 0 (1), c 1 (1) = d 1 (1); linear interpolation can be used to interpolate between c 0 and c 1 , that is.

Beyond the home domain. No canonical parent is asserted for Coons patch. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples

Canonical

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) s t. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → 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; recognition evidence → 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

Applied / In Practice

C(s,t)=L_c(s,t)+L_d(s,t)-B(s,t). The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → A bilinearly blended Coons patch is the surface; invariant → 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; boundary → the case exits the class when 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

Structural Tensions

T1 — Stable identity versus admissible variation. 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. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. 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) s t. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. C(s,t)=L_c(s,t)+L_d(s,t)-B(s,t). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. 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. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. 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) s t. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Coons patch literally, co-instantiate Classification, or only resemble it?

T6 — Autonomy versus reduction. C(s,t)=L_c(s,t)+L_d(s,t)-B(s,t). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Coons patch distinguish that the broader parent Classification leaves together?

Structural–Framed Character

Coons patch is structural-leaning. Its structural side is the repeatable organization summarized by 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. Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: 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. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Classification. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent

What is skeletal. 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. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: B(s,t) = c0(0) (1-s)(1-t) + c0(1) s(1-t) + c1(0) (1-s)t + c1(1) s t. C(s,t)=Lc(s,t)+Ld(s,t)-B(s,t). It further constrains recognition and variation through: 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.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Coons patch literal. Its documented scope includes the condition that 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. Another bounded application condition is that 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 1 (0), c 1 (0) = d 0 (1), c 1 (1) = d 1 (1); linear interpolation can be used to interpolate between c 0 and c 1 , that is. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—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 1 (0), c 1 (0) = d 0 (1), c 1 (1) = d 1 (1); linear interpolation can be used to interpolate between c 0 and c 1 , that is.—and future graph densification may discover a defensible relation only if it preserves that boundary.

  • Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Coons patch. The reviewed identity 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 into elements. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
  • Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

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

Not to Be Confused With

  • Classification. The parent omits the specialist differentia. Tell: Can the case establish 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?
  • Subdivision surface. A smooth limit surface generated by repeatedly refining a coarse polygonal control mesh with a fixed local subdivision rule. Tell: Which entry's carrier, operation, and failure condition are satisfied?
  • Curvelet Transform. A directional multiscale transform with parabolically scaled, increasingly elongated fine-scale elements that sparsely represent smooth curves and curved singularities. Tell: Which entry's carrier, operation, and failure condition are satisfied?
  • Semi-s-cobordism. A cobordism whose inclusion of one designated boundary component is a simple homotopy equivalence, with no corresponding requirement on the other boundary. Tell: Which entry's carrier, operation, and failure condition are satisfied?
  • A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Coons patch remain present if the detector or downstream effect changed?
  • A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Classification?

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Coons_patch (revision 1360118054).
  • Preserved source candidate: https://hdl.handle.net/1721.1/149362
  • Preserved source candidate: http://www.math.hmc.edu/~gu/math142/mellon/Application_to_CAGD/Surface_Construction_Schem.html
  • Preserved source candidate: https://web.archive.org/web/20100806222407/http://www.math.hmc.edu/~gu/math142/mellon/Application_to_CAGD/Surface_Construction_Schem.html

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.