平面代数曲线的排列计算
Arrangement Computation for Planar Algebraic Curves
摘要：
提出了一种计算实平面代数曲线排列的新的证明和完备算法。我们的算法提供了一个几何拓扑分析的分解的平面引起的有限数量的代数曲线在一个柱形代数分解的平面。与以前的方法相比，我们在两个主要方面进行了改进：首先，我们大大减少了精确运算的数量，即我们的算法只使用结果和gcd作为纯符号运算。其次，我们引入了一个新的混合方法，在我们的算法的提升步骤，结合使用认证的数值复根求解器和所得计算的信息。此外，我们从不考虑任何坐标变换，输出也是相对于初始坐标系给出的。我们实现了我们的算法作为一个原型包的C++库CGAL。我们的实现利用图形硬件来加快结果和gcd计算。我们还比较了我们的实现与目前的参考实现，即CGAL的曲线分析和安排代数曲线。对于各种具有挑战性的情况下，我们的实验表明，新的实现优于现有的。
英文摘要：
We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly reduce the amount of exact operations, that is, our algorithms only uses resultant and gcd as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the usage of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system. We implemented our algorithm as a prototypical package of the C++-library CGAL. Our implementation exploits graphics hardware to expedite the resultant and gcd computation. We also compared our implementation with the current reference implementation, that is, CGAL's curve analysis and arrangement for algebraic curves. For various series of challenging instances, our experiments show that the new implementation outperforms the existing one.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
