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    1 #!F-adobe-helvetica-medium-r-normal--18*
    2 #!N 
    3 #!CSeaGreen #!N  #!Rall191 Connections and Interpolation 
    4 #!N #!EC #!N #!N In the cases just discussed, we made 
    5 the implicit assumption that there is a logical connectivity between adjacent 
    6 members of our 2-dimensional or 3-dimensional grid positions. The path connecting 
    7 grid positions is called a  #!F-adobe-times-medium-i-normal--18*   connection #!EF in Data Explorer. 
    8 For a surface (2- or 3-dimensional positions connected by 2-dimensional connections), 
    9 we could choose to make triangular or quadrilateral connections (i.e.,  #!F-adobe-times-medium-i-normal--18*   
   10 triangles #!EF or  #!F-adobe-times-medium-i-normal--18*   quads #!EF ). Quads require four positions 
   11 for each connection and triangles three. Data Explorer supports these  #!F-adobe-times-medium-i-normal--18*   
   12 element types #!EF as well as cubes, tetrahedra, and lines. #!N 
   13 #!N Suppose we first choose to link adjacent positions in the 
   14 botanist's sample area with  #!F-adobe-times-medium-i-normal--18*   line #!EF connections. The grid markers 
   15 were 1 meter on a side. Given a sampling area of 
   16 5 meters by 3 meters, the entire sample would be 15 
   17 meters square; there would be 24 positions (6 in X, and 
   18 4 in Y). On such a plot, we see that a 
   19 position located at [x=0,y=0] is connected to its neighbor at [x=1,y=0]. 
   20 We can imagine that it is meaningful to draw associations between 
   21 data values at adjacent grid positions considering that so many natural 
   22 phenomena are continuous rather than discrete. We assume that the grasses 
   23 are free to spread across the area and the wind is 
   24 free to blow in any direction over the area. #!N #!N 
   25 Previously, we assumed that samples were measured at the center of 
   26 each grid square; that is, the botanist used  #!F-adobe-times-medium-i-normal--18*   quad #!EF 
   27 connections to associate sets of four positions into 4-sided elements, then 
   28 measured data values at the center of each connection element, yielding 
   29 connection-dependent data. Now, assume that the botanist measures temperature values at 
   30 each grid  #!F-adobe-times-medium-i-normal--18*   position #!EF . Temperature would then be position-dependent 
   31 data. It's perfectly acceptable to have both kinds of data in 
   32 the same data set. We will see how this works when 
   33 we discuss  #!F-adobe-times-medium-i-normal--18*   Fields #!EF . #!N #!N Assume that the 
   34 first grid position (sampling point) lies precisely at the position coordinate 
   35 [x=0,y=0]. We take a measurement and record the value. Then we 
   36 measure the temperature at [x=1,y=0]. Later, we ask, what was the 
   37 temperature at [x=0.5,y=0]? Quite honestly, we do not know, because our 
   38 sampling resolution was not fine enough for us to give a 
   39 definitive answer. However, if we make the assumption (very often, a 
   40 perfectly reasonable assumption, but not always!) that our grid overlaid a 
   41 continuous set of values, we can derive the expected data value 
   42 by interpolation between known values. If we use  #!F-adobe-times-medium-i-normal--18*   line #!EF 
   43 connections to connect adjacent points, we realize by looking at our 
   44 mesh that a straight line connects the grid point [x=0,y=0] and 
   45 [x=1,y=0] and that halfway along this line lies the grid point 
   46 [x=0.5,y=0]. We can further assume that the data value at this 
   47 midpoint is the average of the data values at known sample 
   48 points bordering this location. By linear interpolation, we calculate a reasonable 
   49 value for the temperature at [x=0.5,y=0]. #!N #!N We need to 
   50 define polygonal connections over the 2-D grid if we wish to 
   51 find the value at the point [x=0.2,y=0.7]. With  #!F-adobe-times-medium-i-normal--18*   line #!EF 
   52 connections between adjacent pairs of grid points, we can only reasonably 
   53 perform interpolations along those linear boundaries but not into the middle 
   54 of our grid elements. By defining areas bounded by three or 
   55 more points, we can perform interpolation across the area (the polygon 
   56 surface) using weighting functions that take into account the data values 
   57 at all points surrounding the area. In fact, this is the 
   58 same process used by an image-rendering program: it interpolates from known 
   59 values (at the vertices) across the faces of polygons and computes 
   60 the appropriate color at all visible points on the surface, at 
   61 the resolution allowed by the output device (digital file, computer monitor, 
   62 etc.). #!N #!N #!N  #!F-adobe-times-medium-i-normal--18*   Next Topic #!EF #!N #!N  #!Lall192,dxall193 h Identifying Connections  #!EL  
   63 #!N  #!F-adobe-times-medium-i-normal--18*   #!N