Coons patch
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In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
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 w ...
applications, particularly in
finite element method The 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 ...
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, el ...
, 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 graphical methods. He was a professor at the Massachusetts Institute of Technology in the Mechanical Engineering Department. He was also a professor ...
, and date to 1967.Steven A. Coons, Surfaces for computer-aided design of space forms, Technical Report MAC-TR-41, Project MAC, MIT, June 1967.


Bilinear blending

Given four
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...
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 poi ...
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 surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, t ...
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 ...
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 the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
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 correspondi ...
s with the weights chosen to match the partial derivatives at the corners. This forms a bicubically blended Coons patch.


See also

*
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 ...
*
Atlas (topology) In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
* Interpolation


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 , accessdate= 8 April 2012 Multivariate interpolation Splines (mathematics)