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