Triangle based interpolation
 D. F. Watson,
 G. M. Philip
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Triangle based interpolation is introduced by an outline of two classical planar interpolation methods, viz. linear triangular facets and proximal polygons. These are shown to have opposite local bias. By applying cross products of triangles to obtain local gradients, a method designated “slanttop proximal polygon interpolation” is introduced that is intermediate between linear facets and polygonal interpolation in its local bias. This surface is not continuous, but, by extending and weighting the gradient planes, a C^{1} surface can be obtained. The gradients also allow a roughness index to be calculated for each data point in the set. This index is used to control the shape of a blending function that provides a weighted combination of the gradient planes and linear interpolation. This results in a curvilinear, C^{1},interpolation of the data set that is bounded by the linear interpolation and the weighted gradient planes and is tangent to the slanttop interpolation at the data points. These procedures may be applied to data with two, three, or four independent variables.
 Barnhill, R. E. and Farin, G., 1981,C ^{1} Quintic interpolation over triangles—two explicit representations: Int. J. Numer. Meth. Eng., v. 17, p. 1763–1778.
 Bengtsson, B. E. and Nordbeck, S., 1964, Construction of isarithms and isarithmic maps by computers: Bit, v. 4, p. 87–105.
 Bideaux, R. A., 1979, Drill hole management and display,in, Weiss, A. (Ed.), Computer methods for the 80's in the mineral industry: American Institute of Mining, Metallurgical and Petroleum Engineers, Inc., New York, p. 155–162.
 Davis, J. C., 1973, Statistics and data analysis in geology: John Wiley & Sons, New York, 550 pp.
 Franke, R., 1982a, Smooth interpolation of scattered data by local thin plate splines: Comput. Math. Appl., v. 8, p. 273–281.
 Franke, R., 1982b, Scattered data interpolation—tests of some methods: Math. Comput., v. 38, p. 181–200.
 Franke, R. and Nielson, 1980, Smooth interpolation of large sets of scattered data: Int. J. Numer. Meth. Eng., v. 15, p. 1691–1704.
 Gold, C. M., Charters, T. D., and Ramsden, J., 1977, Automated contour mapping using triangular element data structures and an interpolant over each irregular triangular domain: Comput. Graph. (Proceedings of SIGGRAPH '77), v. 11, p. 170–175.
 Klucewicz, I. M., 1978, A piecewiseC ^{1} interpolant to arbitrarily spaced data: Comput. Graph. Image Proc., v. 8, p. 92–112.
 McCullagh, M. J., 1981, Creation of smooth contours over irregularly distributed data using local surface patches: Geograph. Anal., v. 13, p. 51–63.
 Peucker, T. K., 1980, The impact of different mathematical approaches to contouring: Cartographica, v. 17, p. 73–95.
 Philip, G. M. and Watson, D. F., 1982, A precise method for determining contoured surfaces: Austral. Pet. Explor. Assoc. Jour., v. 22, p. 205–212.
 Philip, G. M. and Watson, D. F., 1985. A method for assessing local variation among scattered measurements: (under submission).
 Reedman, J. H., 1979, Techniques in mineral exploration: Applied Science Publishers Ltd., London, 533 pp.
 Rhynsburger, D., 1973, Analytic delineation of Thiessen polygons: Geog. Anal, v. 5, p. 133–144.
 Ripley, B. D., 1981, Spatial statistics: John Wiley & Sons, New York, 252 pp.
 Sabin, M. A., 1980, Contouring—A review of methods for scattered data,in Brodlie, K. W. (Ed.), Mathematical methods in computer graphics and design: Academic Press, New York, p. 63–86.
 Schagen, I. P., 1979, Interpolation in two dimensions—A new technique: Jour. Inst. Math. Appl., v. 23, p. 53–59.
 Schagen, I. P., 1982, Automatic contouring from scattered data points: Comput. Jour., v. 25, p. 7–11.
 Sibson, R., 1981, A brief description of natural neighbour interpolation,in Barnett, V. (Ed.), Interpreting multivariate data: John Wiley & Sons, New York, p. 21–36.
 Watson, D. F., 1981, Computing thendimensional Delaunay tessellation with application to Voronoi polytopes: Comput. Jour., v. 24, p. 167–172.
 Watson, D. F., 1982, ACORD—Automatic contouring of raw data: Comput. Geosci., v. 8, p. 97–101.
 Watson, D. F., 1983, Two images for three dimensions: Pract. Comput., Aug., p. 104–107, Errata, Sept. p. 8.
 Watson, D. F. and Philip, G. M., 1984. Systematic triangulations: Comput. Vis. Graph. Image Proc. v. 26, p. 217–223.
 Watson, D. F. and Philip, G. M., 1985a, The resolution, precision and integration of interpolation: (in preparation).
 Watson, D. F. and Philip, G. M., 1985b, Inverse distance weighted gradients on a triangular network: (under submission).
 Title
 Triangle based interpolation
 Journal

Journal of the International Association for Mathematical Geology
Volume 16, Issue 8 , pp 779795
 Cover Date
 19841101
 DOI
 10.1007/BF01036704
 Print ISSN
 00205958
 Online ISSN
 15738868
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 data analysis
 triangulation
 Delaunay
 Voronoi
 contouring
 scattered data
 linear interpolation
 polygonal interpolation
 curvilinear interpolation
 gradient
 Industry Sectors
 Authors

 D. F. Watson ^{(1)}
 G. M. Philip ^{(1)}
 Author Affiliations

 1. Dept. of Geology and Geophysics, University of Sydney, 2006, Sydney, NSW, Australia