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           Hyperplane Interpolation               

Hyperplane Interpolation

The interpolation method used in Designer smoothly transitions between regions dominated by different data points. To control how this transition is performed, you can choose to do the interpolation with or with out hyperplanes. Using hyperplanes, the algorithm generates an interpolation surface that matches first-order derivatives near the original data points. Without hyperplanes, a flat region is introduced around each of the original data points.

The benefit of using hyperplanes in the interpolation is that the results will more accurately match “smoothly-varying” original data, and will avoid the flat regions introduced by interpolation without hyperplanes. In particular, planar data (of any dimension) will be exactly matched by interpolation with hyperplanes. The three graphs below demonstrate the effect of using hyperplanes.

There are cases when using hyperplanes will lead to less desirable solutions that interpolating without hyperplanes. Using hyperplanes spreads the influence of original data points into neighboring regions. This property can be particularly troublesome when the original data is discontinuous. In interpolation without hyperplanes, an original data point has no influence except in regions that contain the data point. The three graphs below demonstrate this property. In this example, the original data contains a spike. Using hyperplanes leads the interpolated solution to match the average derivative at the second data point, which results in the graph going negative (not what a user would typically desire). The interpolation curve in the second graph (in the region between the second and third original data points) is affected by the location of the first original data point when interpolating with hyperplanes. In interpolation without hyperplanes (third graph) the first original data point has no influence on the interpolation curve in the second region.

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