Surface representations reproducing given digitized contour lines

Abstract

In the practice of administrative or engineering geosciences, the problem of deriving a digital surface representation from a map displaying contour lines of the interesting quantity is quite often encountered. First, alpha-numerical data are retrieved from the map by digitizing these contour lines pointwise into polygons. However, common “gridding” algorithms are known to fail at adequately reproducing the input contour lines due to the inhomogeneous and anisotropic areal distribution of the sites of the data sampled from given contour lines. Therefore, we suggest a new algorithm; the basic elements of its first stage are a constrained Delaunay triangulation of the data sites honoring their natural neighborhood relationship—i.e., whether they belong to the same contour line or not, and linear interpolation according to this triangulation of the data domain. In a second stage, a Bezier-Bernstein or simplex B-spline representation is easily achieved if a C₁ or C₂ smooth representation is required. At this stage, also, discontinuities of the function or its first directional derivatives with known locations in the data domain may be represented, provided this additional information has been taken into account when the triangulation was performed. The algorithm is numerically stable and efficient, and allows external interaction by the user to introduce his/her additional knowledge of the phenomenon to be studied, which may not be explicitly inherent in the available data.