mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Coons patch, is a type of surface patch or

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

parametrization used in

computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...

to smoothly join other

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

s together, and in

computational mechanics Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science (also called scientific computing) as a "third wa ...

applications, particularly in

finite element method Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...

and

boundary element method The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, ele ...

, to mesh problem domains into elements. Coons patches are named after

Steven Anson Coons Steven Anson Coons (March 7, 1912 – August 1979) was an early pioneer in the field of computer graphics methods. He was a professor at the Massachusetts Institute of Technology (MIT) in the mechanical engineering department. After leaving MIT, he ...

, and date to 1967.

Bilinear blending

Given four

space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...

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 In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known po ...

can be used to interpolate between ''c''₀ and ''c''₁, that is :

L_c(s,t)=(1-t) c_0(s)+ t c_1(s)

and between ''d''₀, ''d''₁ :

L_d(s,t)=(1-s) d_0(t)+ s d_1(t)

producing two

ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...

s defined on the unit square. The

bilinear interpolation In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., ''x'' and ''y'') using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be ge ...

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.

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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

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 spline In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the correspondin ...

s with the weights chosen to match the partial derivatives at the corners. This forms a bicubically blended Coons patch.

References

{{cite web, title=Surface Construction Schemes , author=Weiqing Gu , url=http://www.math.hmc.edu/~gu/math142/mellon/Application_to_CAGD/Surface_Construction_Schem.html , archive-url=https://web.archive.org/web/20100806222407/http://www.math.hmc.edu/~gu/math142/mellon/Application_to_CAGD/Surface_Construction_Schem.html , accessdate= 6 August 2010, archive-date=2010-08-06 Multivariate interpolation Splines (mathematics)

Bilinear blending

Bicubic blending

See also

References