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For line features, there are two primary strategies for VtoR conversion: fat and thin. Fat identifies every grid cell cut by a line segment, even if it's just a nick at the corner. Thin identifies the smallest possible set of adjoining cells to characterize a line feature. Although the "thin" option produces pleasing visu- alizations of a line, it discards valuable information. For example, if the line feature was of a stream, then water-law rights/responsibilities are applicable to all cells with a "stream running through it," regardless of whether it's through the center or just in a corner. The upper-right inset in Figure 2 illustrates the conversion of adjacent polygons. The "fat" edge containing the polygon boundary is identified, and geometry is used to determine which polygon is mostly contained within each edge cell. All the interior cells of a polygon are identified to finish the polygon conversion. Figure 2's lower portion depicts approaches for converting raster to vector. For single-cell locations, the x,y coordinates of a cell's centroid are used to position a point feature, and the cell value is used to populate the attribute table. On rare occasions where the cell value indicates the number of points contained in a cell, a random number generator is used to derive coordinates within the cell's geographic extent for the set of points. For gridded line features, the x,y coordinates of the centroids (thin) are frequently used to define the line segments. The set of points often are condensed as appropriate, and a smoothing equa- tion is applied to eliminate the saw-tooth "jaggies." A radically different approach converts the sides of the cells into a thin polygon capturing the area of possible inclusion of the grid-based line—sort of a narrow corridor for the line. For gridded polygons, the sides at the edges of abutting cells are used to define the feature's boundary line, its points are condensed and smoothed, and the adjoining polygons' topology is added. For isolated gridded polygons, the outside edges commonly are used to identify the polygon's boundary line. Interpolate and Slice Figure 3 deals with converting continuous map sur- faces to discrete vector representations. A frequently used technique that generates true contour lines of constant value was described in detail in last month's column. The procedure involves identifying cell values that bracket a desired contour level, then interpolating the x,y coordinates for points between all the cell-value pairs and connecting the new points for a line of constant value. The black lines in Figure 3 identify the set of interpolated 200-foot contours lines for a dataset with values from 500 to 2,500. Another commonly used technique involves slicing the data range into a desired number of contour intervals. For example, 2,500 - 500 / 10 = 200 identi- fies the data step used in generating the data ranges of the figure's contour intervals (color bands). The first contour range from 500 to 700 "color-fills" with red all grid-cell values within this range, orange for values 700 to 900, tan for values 900 to 1,100, etc. The 3-D surface shows the contour-interval classification draped over actual data values stored in the grid. The added red lines in the enlarged inset identify the edges of the grid-cell contour-interval groupings. In the enlarged 3-D plot, there are numerous differ- ing data values, as the red border goes up and down Figure 2. Basic procedures demonstrate direct calculation-based vector-to-raster (VtoR) conversion and the reverse (RtoV). with data values along the sawtooth edge. As previ- ously noted, this boundary can be smoothed (dotted red) and used for the borders of the contour-interval polygons generated in the RtoV conversion. The bottom line (pun intended) is that in many mapped data visualizations, the boundary (border) out- lining a contour interval isn't the same as a contour line (of constant value). That's the beauty of grid vs. vector data structures—they are not the same. Author's Notes: For more information on grid-based mapped data considerations, see the online book, Beyond Model- ing III, Topic 18, Understanding Grid-based Data, at www. innovativegis.com/Basis/MapAnalysis. Figure 3. Maps compare the basic approaches for identifying contour lines and contour-interval boundaries. DECEMBER 2O11 / WWW . GEOPLA CE . C O M 11