**Automated contour mapping using triangular element data structures and an interpolant over each irregular triangular domain****Author(s): **C. M. Gold, T.D. Charters, J. Ramsden.**Proceedings: **ACM SIGGRAPH Computer Graphics, Vol. 11, No. 2, pp. 170--175,
1977.

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**Abstract: **

Surface display techniques, including contour mapping, usually use an interpolant
when data values are on a rectangular grid; but when values are irregularly located some
form of assumed or estimated distance-weighting function is often used to estimate the
"influence" of data points at other locations. If moving average (or some other)
techniques are then applied, slopes may be zero at data points; extrema may fall at data
points exclusively; or surfaces may not pass through data points.
An alternative method is proposed whereby a map area is automatically divided into
suitable triangular domains with a data point at each vertex. Surface estimation and
contour plotting are then performed independently for each triangle using a measured or
best-fit plane associated with each data point. The approach requires the utilization of
three non-original aspects: 1) the derivation of a local homogeneous "area" coordinate
system for any arbitrary triangle; 2) the construction of a data-structure linking each
triangular domain with its three neighbouring triangles and three associated data points,
and 3) the use of a "conforming" triangular finite-element interpolating function.
Use of the first two of these concepts permits the economic generation and
optimization of a triangular mesh from a set of data points. Optimization criteria used to
define the "best" triangular partition are described in some detail, along with computer
timing for this step. Use of the first and third concepts permits the interpolation of a
smooth surface over the whole map area even though each triangular element is estimated
and plotted independently. The requirements for a suitable interpolating function are
discussed, and an interpolant is suggested that preserves elevation and slope at each data
point as well as elevation and slope continuity between domains. Extrema need not be
located at data points but are constrained by their associated planes. Contour line
segments are produced by division of each triangle into N2 sub-triangles, elevation
estimation at each resulting node, and linear interpolation and plotting withing each subtriangle. Resolution is then a function of N and local data point density. Extension of
the method to higher dimensions is briefly discussed.

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