![]() The circles illustrate how the circle circumscribing each triangle has no other points from the set within it. Many programs support the creation of the triangulation required, for example the delaunay( x, y) function in MATLab where x, and y are n x1 vectors containing the coordinates of all the points, or (as here) the Grid Data operation in Surfer, using the option to export the triangulation used as a base map. Each location is represented by an ( x,y) pair. Figure 4‑32, below, illustrates this process for a set of spot height locations in a small area south of Edinburgh in Scotland (GB Ordnance Survey tile NT04). This states that three points form a Delaunay Triangulation if and only if ( iff) a circle which passes through all three points contains no other points in the set. To ensure the triangulation will have the best chance of meeting the characteristics desired, a construction rule was devised by the mathematician Delaunay. It has also been found that a desirable characteristic of such triangulations is that long thin triangles with very acute internal angles are to be avoided in order to provide the best framework for measurement and analysis. It has long been known that triangulation provides a secure method for locating points on the Earth’s surface by field survey. Given a set of points in the plane, P, lines may be drawn between these points to create a complete set of non-overlapping triangles (a triangulation) with the outer boundary being the convex hull of the point set. GIS does make extensive use of irregular triangular tessellations, both for division of plane regions and as an efficient means of representing surfaces. Regular triangular and hexagonal grids are also possible in the plane but are rarely implemented in software packages. Within GIS regular tiles are almost always either square or rectangular, and form a (continuous) grid structure. As we noted in Table 1‑1 a (regular or irregular) tessellation of a plane involves the subdivision of the plane into polygonal tiles that completely cover it. A region may be divided into a set of non-overlapping areas in many ways.
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