具有多项式密度分布的直线骨架卷积曲面

给出了一个具有多项式密度分布的直线骨架卷积曲面的解析表达式，并提出了基于控制曲线的密度控制方法，实验结果表明，该方法的自然景物，海生物等光滑物体造型中具有很大的应用价值。

Convolution Surface Modeling Based on Line Segment Primitive with Polynomial Density Distribution

Convolution surface is a natural and powerful extension of point based metaball surfaces, and provides a powerful and flexible representation for modeling objects with highly complex topology, such as plants, sea life forms and objects whose topology changes over time. Therefore it is of important use in the modeling of natural phenomenon and computer animation. Convolution surface is obtained by convolving the skeleton with a three dimensional low pass filter kernel. The skeletal elements in convolution surfaces may be points, line segments, curves, polygons and other geometrical modeling primitives. The line segment primitive can be considered as one of the most fundamental ones since many objects can be abstracted as curve skeletons, and curve skeletons can in turn be subdivided into line segments. The analytical model for line segment primitive derived by McCormack and Sherstyuk treats the density distribution along the skeleton uniformly, thus their model can only coat the skeletal line segment with a convolution surface of a constant radius. This restriction means that modeling a surface of varying radius is impossible. Such non constant radius and tapering shapes are prevalent in organic objects, most notably in botanical stems and branches. To overcome this problem, this paper proposes a closed form model for controlling line skeletons with polynomial density distributions by adopting Cauchy kernel function. By applying integration techniques, the analytical field formulae with three lowest polynomial distributions are given in the paper. The analytical formulae for higher degree polynomial distribution can be deduced similarly. The density distribution along a skeleton can be intuitively specified by a cubic control spline curve. Since any curve skeleton can be approximated by line segments, our method is also applicable to any curve abstracted objects. The computation requirement for polynomial density distribution is competitive compared with the case of uniform density distribution. By reusing calculated results, the incremental cost from constant distribution to linear distribution, which is most often adopted, is very little. Only additional 5 multiplications/divisions and 2 additions/subtractions are needed. Even for the cubic density distribution, the increase in the cost is less than double.