Triangular Networks

Triangular Networks

Triangular networks are defined as grids of triangle or tetrahedron cells where all of the nodes in the grid are exclusively those in the sample data. For these types of grids, the cell connectivity must be explicitly defined. In two dimensions, these grids are referred to as Triangulated Irregular Networks or TINs. The 3D equivalent grids are Tetrahedral Irregular Networks.

Triangulated Irregular Networks – 2D

Delaunay triangulation is one of the most commonly used methods for creating TINs. By definition, 3 points form a Delaunay triangle if and only if the circle defined by them contains no other point. Focusing on creating Delaunay triangles produces triangles with fat (large) angles that have preferred rendering characteristics. The boundary edges on the Delaunay network form the convex hull, which is the smallest area convex polygon to contain all of the vertices.

Figure 0.17 Flat-Shaded Delaunay TIN of Geologic Surface

The TIN surface above (Figure 1.18) has significant variation in the size of the triangles. This is a natural consequence of the grid's being created using only nodes from the input data file. When such a surface is rendered with data, having very large triangles can result in very objectionable visualization anomalies. These anomalies result from rendering large triangles that have a range of data values that span a significant fraction of the total data range. There are many methods that could be used to assign color to each triangle. These methods are referred to as surface rendering modes.

Two of the most commonly used rendering modes are flat shading and Gouraud shading. Flat shading assigns a single color to the entire triangle. The color is computed based on the average elevation (data value) for that triangle, lighting parameters and orientation to the viewer camera. In the upper left corner we have a large single triangle that spans a significant range of elevations. When it is assigned a color that corresponds to the mean elevation for that triangle, that color will be wrong. More precisely, the color does not fall within the color scale. Note the color of the triangle in the upper right corner of Figure 1.18 and the one below it. The color of these triangles is outside the range of our color scale.

The problem of large triangles is no better when using Gouraud shading. Gouraud shading assigns colors to each node of the triangle based on the data values. This assures that the colors at the nodes (vertices of the triangles) will be correct. Colors are then interpolated over the area of the triangle based on lighting parameters and orientation to the viewer camera. Consider the triangle in the upper right hand corner of Figure 1.19. The upper right node is assigned the color blue (corresponding to a low value) and the upper left node is assigned the color red (corresponding to a high value). The color scale for this problem ranges from blue to cyan to green to yellow to red. However, for this anomalous situation the color that will be interpolated between blue and red along the uppermost edge will be magenta. Magenta is not a color in our range of colors.

Figure 0.18 Gouraud-Shaded Delaunay TIN of Geologic Surface

To overcome the problems caused by large triangles, the triangles can be refined (subdivided) to create a grid that still contains points that honor the original input nodes, but has more uniform cell sizes. In Figure 1.20 (which has a spatial extent of 500 feet in x and 380 feet in y) it was specified that no triangle's edge may exceed 45 feet in length. We must interpolate the elevation values (or our data values) to these new nodes created as a result of the triangle subdivision. The simplest means of doing this is bilinear interpolation. The refined TIN grid with bilinear interpolation and flat shaded triangles is shown in Figure 1.21. Note that the all of the triangles have appropriate colors. To avoid the large cell coloring problem (this is a problem with all cell types except points), no single cell should have data values at its nodes that span more than about 20 percent of the total data range.

Figure 0.19 Flat-Shaded Subdivided TIN of Geologic Surface

If Gouraud shading is employed instead of flat shading, the resultant surface has a smoother appearance, however the fundamental linear interpolation along cell edges is still evident in the colors. If the maximum triangle size were made much smaller, the flat shaded model would approach the appearance of the Gouraud shaded model. However, without using a different interpolation approach the Gouraud-shaded model would not change dramatically.

Figure 0.20 Gouraud-Shaded Subdivided TIN of Geologic Surface

EVS includes another technique for coloring surfaces. This method, called solid contours, assigns uniform color bands based on the data values. Figure 1.22 demonstrates this method that subdivides cells using bilinear interpolation. Because this method inherently includes triangle subdivision using bilinear interpolation, the figure would be identical whether the input grid was the large triangles from the original TIN surface or the refined smaller triangles. The boundaries of the colored bands are effectively isopachs (isolines) of constant elevation.

Figure 0.21 Solid Contour TIN of Geologic Surface

To complete this discussion and comparison of gridding and interpolation methods, the same data file was used to create a convex hull grid and the elevation data was estimated using EVS's two-dimensional kriging software. Kriging will be discussed in more detail in section 1.3.3. This technique honors all of the original data points, but creates much smoother distributions between the values. The result shown in Figure 1.23 is a more realistic and aesthetically superior surface. Labeled isolines on 10 foot intervals were added to this figure. Note that these isolines are similar, but much smoother than those in Figure 1.22.

Figure 0.22 Kriged 2D Convex Hull Grid

Tetrahedral Irregular Networks – 3D

Tetrahedral Irregular Networks provide a method to create a volumetric representation of a three-dimensional set of points. As with a TIN, the nodes in the resulting grid are exclusively those in the original measured sample data. Tetrahedral Irregular Networks use tetrahedron cells to fill the three-dimensional convex hull of the data as shown in Figure 1.24. The result often contains cells of widely varying volumes having potentially large data variation across individual cells. For this and other reasons, this approach is not often used.

Figure 0.23 Tetrahedral Irregular Network

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