We consider the problem of computing the topology and describing the geometry of a parametric curve in $\mathbb{R}^n$. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter space and not in the implicit space. When the parametrization involves polynomials of degree at most $d$ and maximum bitsize of coefficients $\tau$, then the worst case bit complexity of PTOPO is $ \widetilde{\mathcal{O}}_B(nd^6+nd^5\tau+d^4(n^2+n\tau)+d^3(n^2\tau+ n^3)+n^3d^2\tau)$. This bound matches the current record bound $\widetilde{\mathcal{O}}_B(d^6+d^5\tau)$ for the problem of computing the topology of a plane algebraic curve given in implicit form. For plane and space curves, if $N = \max\{d, \tau \}$, the complexity of PTOPO becomes $\widetilde{\mathcal{O}}_B(N^6)$, which improves the state-of-the-art result, due to Alc\'azar and D\'iaz-Toca [CAGD'10], by a factor of $N^{10}$. In the same time complexity, we obtain a graph whose straight-line embedding is isotopic to the curve. However, visualizing the curve on top of the abstract graph construction, increases the bound to $\widetilde{\mathcal{O}}_B(N^7)$. For curves of general dimension, we can also distinguish between ordinary and non-ordinary real singularities and determine their multiplicities in the same expected complexity of PTOPO by employing the algorithm of Blasco and P\'erez-D\'iaz [CAGD'19]. We have implemented PTOPO in Maple for the case of plane and space curves. Our experiments illustrate its practical nature.

中文翻译：

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PTOPO：计算参数曲线的几何和拓扑

我们考虑在$ \ mathbb {R} ^ n $中计算拓扑并描述参数曲线的几何形状的问题。我们提出一种算法PTOPO，该算法构造一个抽象图，该抽象图与嵌入空间中的曲线是同位的。我们的方法利用了参数表示的优点，并且没有求助于隐式化。最重要的是，我们在参数空间而不是隐式空间中执行所有计算。当参数化涉及度为$ d $的多项式和系数$ \ tau $的最大位大小时，PTOPO的最坏情况的位复杂度为$ \ widetilde {\ mathcal {O}} _ B（nd ^ 6 + nd ^ 5 \ tau + d ^ 4（n ^ 2 + n \ tau）+ d ^ 3（n ^ 2 \ tau + n ^ 3）+ n ^ 3d ^ 2 \ tau）$。此边界与当前记录边界$ \ widetilde {\ mathcal {O}} _ B（d ^ 6 + d ^ 5 \ tau）$匹配，用于计算以隐式形式给出的平面代数曲线的拓扑问题。对于平面和空间曲线，如果$ N = \ max \ {d，\ tau \} $，PTOPO的复杂度变为$ \ widetilde {\ mathcal {O}} _ B（N ^ 6）$，从而改善了状态-由于Alc'azar和D \'iaz-Toca [CAGD'10]导致的最先进的结果是$ N ^ {10} $。在相同的时间复杂度下，我们获得了图，其直线嵌入与曲线的同位素相同。但是，在抽象图构造之上可视化曲线，会增加到$ \ widetilde {\ mathcal {O}} _ B（N ^ 7）$的边界。对于一般尺寸的曲线，我们还可以通过使用Blasco和P'erez-D'iaz [CAGD'19]算法来区分普通和非普通实奇异点，并以相同的PTOPO预期复杂度确定它们的多重性。对于平面和空间曲线，我们已经在Maple中实现了PTOPO。我们的实验说明了它的实用性。