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Constrained fitting with free-form curves and surfaces
Computer-Aided Design ( IF 4.3 ) Pub Date : 2020-01-21 , DOI: 10.1016/j.cad.2020.102816
István Kovács , Tamás Várady

The accurate reconstruction of engineering parts from measured data is a challenging problem, in particular, when various geometric constraints need to be imposed to meet requirements in downstream CAD/CAM applications. There is a wide range of constraints including incidence, tangency, orthogonality, parallelism, symmetry and many others. In the majority of cases, only regular surfaces (planes, natural quadrics, sweeps) are involved; however, imposing constraints on objects with free-form curves and surfaces is also necessary in engineering practice; this motivated our work.

We numerically optimize complex systems of non-linear equations, containing unknown control points and auxiliary geometric entities. We show that auxiliaries are indispensable; their use simplifies the algebra of the constraint equations and speeds up computations. We do not require the auxiliaries to be fixed in advance, rather propose incorporating them into the constraint system. As a result, the procedure will automatically substantiate various entities that have been unknown beforehand, for example, points of intersection or constrained trim curves on B-spline surfaces.

After formulating equations for a representative set of free-form constraints, we will discuss how this technique can be applied in practical reverse engineering. We will present case studies and analyze how surfaces get improved by means of constrained fitting.



中文翻译:

自由曲线和曲面的约束拟合

根据测量数据准确重建工程零件是一个具有挑战性的问题,特别是当需要施加各种几何约束来满足下游CAD / CAM应用程序的要求时。约束范围很广,包括入射,相切,正交,并行,对称和许多其他约束。在大多数情况下,仅涉及规则曲面(平面,自然二次曲面,后掠角);但是,在工程实践中,也必须对具有自由形式的曲线和曲面的对象施加约束。这激励了我们的工作。

我们在数值上优化包含未知控制点和辅助几何实体的非线性方程组的复杂系统。我们证明助剂是必不可少的。它们的使用简化了约束方程的代数并加快了计算速度。我们不要求预先固定辅助设备,而是建议将其合并到约束系统中。结果,该过程将自动证实事先未知的各种实体,例如B样条曲面上的交点或约束修剪曲线。

在为一组代表性的自由形式约束公式表示之后,我们将讨论如何将该技术应用于实际的逆向工程。我们将进行案例研究,并分析如何通过约束拟合来改善曲面。

更新日期:2020-03-07
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