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Indirect Predicates for Geometric Constructions
Computer-Aided Design ( IF 4.3 ) Pub Date : 2020-04-29 , DOI: 10.1016/j.cad.2020.102856
Marco Attene

Geometric predicates are a basic ingredient to implement a vast range of algorithms in computational geometry. Modern implementations employ floating point filtering techniques to combine efficiency and robustness, and state-of-the-art predicates are guaranteed to be always exact while being only slightly slower than corresponding (inexact) floating point implementations. Unfortunately, if the input to these predicates is an intermediate construction of an algorithm, its floating point representation may be affected by an approximation error, and correctness is no longer guaranteed.

This paper introduces the concept of indirect geometric predicate: instead of taking the intermediate construction as an explicit input, an indirect predicate considers the primitive geometric elements which are combined to produce such a construction. This makes it possible to keep track of the floating point approximation, and thus to exploit efficient filters and expansion arithmetic to exactly resolve the predicate with minimal overhead with respect to a naive floating point implementation. As a representative example, we show how to extend standard predicates to the case of points of intersection of linear elements (i.e. lines and planes) and show that, on classical problems, this approach outperforms state-of-the-art solutions based on lazy exact intermediate representations.



中文翻译:

几何构造的间接谓词

几何谓词是在计算几何中实现各种算法的基本要素。现代实现采用浮点过滤技术来结合效率和鲁棒性,并且最先进的谓词始终保持精确,同时仅比相应的(不精确的)浮点实现慢。不幸的是,如果这些谓词的输入是算法的中间结构,则其浮点表示法可能会受到近似误差的影响,并且不再保证正确性。

本文介绍了间接几何谓词的概念:间接谓词不是将中间构造作为显式输入,而是将原始几何元素组合在一起产生这样的构造。这样就可以跟踪浮点近似值,从而可以利用有效的过滤器和扩展算法以相对于朴素的浮点实现以最小的开销准确地解析谓词。作为一个有代表性的例子,我们展示了如何将标准谓词扩展到线性元素(即线和平面)的交点的情况,并表明在经典问题上,该方法的性能优于基于惰性的最新解决方案。确切的中间表示。

更新日期:2020-04-29
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