Surface interpolation scheme by distance blending over convex sets

Interpolation and approximation of different types of data points are very important in the field of CAD/CAM, virtual reality and computer graphics. Besides interpolating sample data, we also need to interpolate sample pieces of surfaces, textures or functions located over different areas or sets. Here we actually intend to interpolate an infinite number of data points of any function or pattern. In this chapter, we introduce the concept of distance from a point to a set or area. The square of distance junction is Cl smooth when the set is convex. The distance function is then used to construct a blending basis for interpolating functions or surfaces defined on related convex sets. An explicit set-splines basis is then derived, which has many useful properties similar to the traditional B-spline basis. These set-splines can be used to not only interpolate sample data, but sample pieces of functions as well. The potential of this new method is great, for instance, blending or smoothing surfaces, interpolating textures and images, and interpolating volume data.