Minimum curvature variation curves, networks, and surfaces for fair free-form shape design
Minimum curvature variation curves, networks, and surfaces for fair free-form shape design
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发表时间：
1993-04
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通讯作者：
Henry P. Moreton
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作者：
Henry P. Moreton
传统的设计自由曲线和曲面的方法侧重于实现特定水平的单元间连续性。这些方法结合了启发式和构造来实现最终的形状。尽管使用这些方法构建的形状在技术上是连续的，但它们已经被证明缺乏公平性，具有不受欢迎的缺陷，如凸起和皱纹。公平性与曲率的平滑和最小变化密切相关。在这项工作中，我们提出了一种曲线和曲面设计的新技术，该技术将基于几何的规范与公平性函数的约束优化（最小化）相结合。通过适当的惩罚函数将单元间连续问题纳入最小化，简单地解决了实现单元间连续的难题。传统的公平措施是基于应变能，我们已经开发了一个更好的公平措施：曲率的变化。除了生产质量明显优越的物体外，曲率的变化最小化使得对规则形状的建模变得微不足道，例如圆和圆环，一类表面包括：球体，圆柱体，锥体和环面。本文介绍了曲率变化作为公平性度量、最小变化曲线（MVC）、最小变化网络（MVN）和最小变化面（MVS）。MVC将曲率的弧长导数的平方的弧长积分最小化，同时插值一组几何约束，包括位置，以及可选的切线方向和曲率。MVN在插值由曲面位置、切平面和曲面曲率组成的几何约束网络时最小化相同的函数。最后，通过跨越MVN的开口来获得MVS，同时最小化测量表面曲率变化的表面函数。我们将介绍上述技术的细节，并描述一些替代方法之间的权衡。给出了求解插值难题的方法，并与传统插值方法进行了比较。两者都证明了曲率变化作为公平度量的优越性和优化作为形状设计工具的有效性，尽管需要大量的计算成本。
Traditionally methods for the design of free-form curves and surfaces focus on achieving a specific level of inter-element continuity. These methods use a combination of heuristics and constructions to achieve an ultimate shape. Though shapes constructed using these methods are technically continuous, they have been shown to lack fairness, possessing undesirable blemishes such as bulges and wrinkles. Fairness is closely related to the smooth and minimal variation of curvature. In this work we present a new technique for curve and surface design that combines a geometrically based specification with constrained optimization (minimization) of a fairness functional. The difficult problem of achieving inter-element continuity is solved simply by incorporating it into the minimization via appropriate penalty functions. Where traditional fairness measures are based on strain energy, we have developed a better measure of fairness: the variation of curvature. In addition to producing objects of clearly superior quality, minimizing the variation of curvature makes it trivial to model regular shapes such as, circles and cyclides, a class of surface including: spheres, cylinders, cones, and tori. In this thesis we introduce: curvature variation as a fairness metric, the minimum variation curve (MVC), the minimum variation network (MVN), and the minimum variation surface (MVS). MVC minimize the arc length integral of the square of the arc length derivative of curvature while interpolating a set of geometric constraints consisting of position, and optionally tangent direction and curvature. MVN minimize the same functional while interpolating a network of geometric constraints consisting of surface position, tangent plane, and surface curvatures. Finally, MVS are obtained by spanning the openings of the MVN while minimizing a surface functional that measures the variation of surface curvature. We present the details of the techniques outlined above and describe the trade-offs between some alternative approaches. Solutions to difficult interpolation problems and comparisons with traditional methods are provided. Both demonstrate the superiority of curvature variation as a fairness metric and efficacy of optimization as a tool in shape design, albeit at significant computational cost.