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 5the implicit assumption that there is a logical connectivity between adjacent 6members of our 2-dimensional or 3-dimensional grid positions. The path connecting 7grid positions is called a #!F-adobe-times-medium-i-normal--18* connection #!EF in Data Explorer. 8For a surface (2- or 3-dimensional positions connected by 2-dimensional connections), 9we could choose to make triangular or quadrilateral connections (i.e., #!F-adobe-times-medium-i-normal--18* 10triangles #!EF or #!F-adobe-times-medium-i-normal--18* quads #!EF ). Quads require four positions 11for each connection and triangles three. Data Explorer supports these #!F-adobe-times-medium-i-normal--18* 12element types #!EF as well as cubes, tetrahedra, and lines. #!N 13#!N Suppose we first choose to link adjacent positions in the 14botanist's sample area with #!F-adobe-times-medium-i-normal--18* line #!EF connections. The grid markers 15were 1 meter on a side. Given a sampling area of 165 meters by 3 meters, the entire sample would be 15 17meters square; there would be 24 positions (6 in X, and 184 in Y). On such a plot, we see that a 19position located at [x=0,y=0] is connected to its neighbor at [x=1,y=0]. 20We can imagine that it is meaningful to draw associations between 21data values at adjacent grid positions considering that so many natural 22phenomena are continuous rather than discrete. We assume that the grasses 23are free to spread across the area and the wind is 24free to blow in any direction over the area. #!N #!N 25Previously, we assumed that samples were measured at the center of 26each grid square; that is, the botanist used #!F-adobe-times-medium-i-normal--18* quad #!EF 27connections to associate sets of four positions into 4-sided elements, then 28measured data values at the center of each connection element, yielding 29connection-dependent data. Now, assume that the botanist measures temperature values at 30each grid #!F-adobe-times-medium-i-normal--18* position #!EF . Temperature would then be position-dependent 31data. It's perfectly acceptable to have both kinds of data in 32the same data set. We will see how this works when 33we discuss #!F-adobe-times-medium-i-normal--18* Fields #!EF . #!N #!N Assume that the 34first 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 36measure the temperature at [x=1,y=0]. Later, we ask, what was the 37temperature at [x=0.5,y=0]? Quite honestly, we do not know, because our 38sampling resolution was not fine enough for us to give a 39definitive answer. However, if we make the assumption (very often, a 40perfectly reasonable assumption, but not always!) that our grid overlaid a 41continuous set of values, we can derive the expected data value 42by interpolation between known values. If we use #!F-adobe-times-medium-i-normal--18* line #!EF 43connections to connect adjacent points, we realize by looking at our 44mesh 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 47midpoint is the average of the data values at known sample 48points bordering this location. By linear interpolation, we calculate a reasonable 49value for the temperature at [x=0.5,y=0]. #!N #!N We need to 50define polygonal connections over the 2-D grid if we wish to 51find the value at the point [x=0.2,y=0.7]. With #!F-adobe-times-medium-i-normal--18* line #!EF 52connections between adjacent pairs of grid points, we can only reasonably 53perform interpolations along those linear boundaries but not into the middle 54of our grid elements. By defining areas bounded by three or 55more points, we can perform interpolation across the area (the polygon 56surface) using weighting functions that take into account the data values 57at all points surrounding the area. In fact, this is the 58same process used by an image-rendering program: it interpolates from known 59values (at the vertices) across the faces of polygons and computes 60the appropriate color at all visible points on the surface, at 61the resolution allowed by the output device (digital file, computer monitor, 62etc.). #!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 64