BLEND
Transfinite Interpolation

BLEND is a C library which "blends" data.

This is a common way of creating a smooth set of data based on a small set of known values. For instance, if we measure the temperature every hour, we naturally assume that the temperature at 2:15 can be approximated by "blending" 1/4 of the temperature at 3:00 and 3/4 of the temperature at 2:00.

Now suppose that we take the temperature at evenly spaced points on the floor of a room. We can again see how to use blending so that, in each little square, we take a blend of the values at the four corners to get the value at any point within the square. Similar ideas can be used with a cube.

More complicated cases might arise where we know the temperature everywhere along lines, or along planes that cut through a cube. Even then, it is possible to blend the data in a smooth and sensible way.

BLEND interpolates values based on a set of given data. BLEND can handle input data that is 1, 2, or 3 dimensional. In the general, 3D case, the data can depend on smoothly varying space parameters (R,S,T) or on tabular indices (I,J,K). The data may be given at the corners, edges, or faces of the unit cube. In the (R,S,T) case, BLEND can supply an interpolated value at any point in the cube. In the (I,J,K) case, BLEND will fill in tabular values for all intermediate indices.

In the simplest case, where BLEND is only given data values at the endpoints of a line segment, the 4 corners of a square, or the 8 corners of a cube, BLEND is equivalent to linear, bilinear or trilinear finite element interpolation of the data. However, in the more interesting cases where BLEND is given, say, a formula for the data along the sides of the square, or the edges of the cube, the interpolation is called "transfinite", since in theory it samples the input data at more than a finite number of points.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

BLEND is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Reference:

1. William Gordon,
   Blending-Function Methods of Bivariate and Multivariate Interpolation and Approximation,
   SIAM Journal on Numerical Analysis,
   Volume 8, Number 1, March 1971, pages 158-177.
2. William Gordon, Charles Hall,
   Transfinite Element Methods: Blending-Function Interpolation over Arbitrary Curved Element Domains,
   Numerische Mathematik,
   Volume 21, Number 1, February 1973, pages 109-129.
3. William Gordon, Charles Hall,
   Construction of Curvilinear Coordinate Systems and Application to Mesh Generation,
   International Journal of Numerical Methods in Engineering,
   Volume 7, 1973, pages 461-477.
4. Charles Hall, Thomas Porsching,
   Numerical Analysis of Partial Differential Equations,
   Prentice-Hall, 1990,
   ISBN: 013626557X,
   LC: QA374.H29.
5. Joe Thompson, Bharat Soni, Nigel Weatherill,
   Handbook of Grid Generation,
   CRC Press, 1999,
   ISBN: 0849326877.

Source Code:

• blend.c, the source code;
• blend.h, the include file;

Examples and Tests:

• blend_test.c, the calling program;
• blend_test.txt, the output file.

List of Routines:

• BLEND_0D1 extends scalar data at endpoints to a line.
• BLEND_1D1 extends scalar data along the boundary into a square.
• BLEND_2D1 extends scalar data along the surface into a cube.
• BLEND_I_0D1 extends indexed scalar data at endpoints along a line.
• BLEND_IJ_0D1 extends indexed scalar data at corners into a table.
• BLEND_IJ_1D1 extends indexed scalar data along edges into a table.
• BLEND_IJK_0D1 extends indexed scalar data along corners into a cubic table.
• BLEND_IJK_1D1 extends indexed scalar data along "edges" into a cubic table.
• BLEND_IJK_2D1 extends indexed scalar data along faces into a cubic table.
• BLEND_R_0DN extends vector data at endpoints into a line.
• BLEND_RS_0DN extends vector data at corners into a square.
• BLEND_RS_1DN extends vector data along sides into a square.
• BLEND_RST_0DN extends vector data at corners into a cube.
• BLEND_RST_1DN extends vector data on edges into a cube.
• BLEND_RST_2DN extends vector data on faces into a cube.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• R8BLOCK_PRINT prints a double precision block (a 3D matrix).
• R8MAT_PRINT prints an R8MAT, with an optional title.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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