Obtuse-Angle Shadowing Networks and Distance-Based Interpolation
 This Demonstration illustrates obtuse-angle shadowing networks and distance-based interpolation; to make it visual, we use 2D points, which you can create (Alt+Click) and drag. Two points A and B in the network are NOT connected if there are two other points C and D such that C is connected to A, D is connected to B, and both the angles ACB and ADB are obtuse. (Another possibility, not shown here, is to use directed connections: A is NOT connected to B if there exists another point C in the point set, such that C is connected to B and the angle ACB is obtuse.) Further, two points in the network are not connected if the distance between them is zero or larger than twice the cutoff radius. The cutoff radius is an optional parameter, which you can vary. For small systems, the cutoff radius can be large or infinite, but for large systems we might get better speed and scaling properties by a smaller choice. With a hierarchical organization of the points, we could then avoid calculation and storage of the complete distance matrix. (That is not included in this Demonstration.) Note the difference between this kind of network and Delaunay triangulation: here some of the points are only connected to one of the other points. Occasionally two connections might cross. We can estimate the dimensionality of the local environment to a given point by looking at the number of connected points. If you look at a random infinite network, you can estimate the number of connected neighbors: The "Generate a new point set!" button generates a whole set of new random points and function values. With the "no interpolation" button selected, you see just the obtuse-angle shadowing network connecting some random points. In this mode the response to changes in the point set is rapid. When one of the interpolation buttons 0, 1, or 2 is selected, you see the interpolation function when random function values are assigned to the network points. You can change the function values one at a time with the two green dot sliders. The selected point is green. To keep updates fast, the settings of the contour plot sometimes show jagged curves. The zeroth-order interpolated function is piecewise constant; it is essentially the average of the surrounding neighbor points. The first-order interpolated function is continuous, but does not have a continuous derivative. The interpolation coefficients are always positive or strictly zero, so the interpolation always gives very "conservative" results, never beyond the control point values. The second-order interpolated function has a continuous derivative, with interpolation coefficients that might be negative. This function is able to suggest positions for minima and maxima. To calculate the second-order interpolation, both the first level and the second level of connected points are used. The second level of connected points is calculated in the same way as the first level, just ignoring all first-level connections. These methods have probably been invented several times before, but I have not found any information about that in the easily available surface layer. Please inform me if you know! The programming is to be seen as "proof-of-principle", and I have tried to keep it simple. The algorithms could surely be improved in several aspects, especially the speed for a large number of points. I find the simplicity and the generality of the methods attractive. One could think of different applications: • Organizing sets of color points, for example, to be able to do dithering with a finite number of colors • Finding topological features of experimental datasets: interior, exterior, and connecting points, clusters, voids, etc.  "Obtuse-Angle Shadowing Networks and Distance-Based Interpolation" from The Wolfram Demonstrations Project http://demonstrations.wolfram.com/ObtuseAngleShadowingNetworksAndDistanceBasedInterpolation/ Contributed by: Ingolf Dahl | ||||||||||||||
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