We consider the problem of computing the topology and describing the geometry of a parametric curve in Rn. 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.

We investigate the arithmetic-geometric structure of the lecture hall cone $L_n \ := \ \big\{\lambda\in \mathbb{R}^n: \, 0\leq \frac{\lambda_1}{1}\leq \frac{\lambda_2}{2}\leq \frac{\lambda_3}{3}\leq \cdots \leq \frac{\lambda_n}{n}\big\}$. We show that $L_n$ is isomorphic to the cone over the lattice pyramid of a reflexive simplex whose Ehrhart $h^*$-polynomial is given by the $(n-1)$st Eulerian polynomial and prove that lecture hall cones admit regular, flag, unimodular triangulations. After explicitly describing the Hilbert basis for $L_n$, we conclude with observations and a conjecture regarding the structure of unimodular triangulations of $L_n$, including connections between enumerative and algebraic properties of $L_n$ and cones over unit cubes.

Based on the computation of a polytope Q, called the predicted polytope, containing the Newton polytope P of the implicit equation, implicitization of a parametric hypersurface is reduced to computing the nullspace of a numeric matrix. Polytope Q may contain P as a Minkowski summand, thus jeopardizing the efficiency of sparse implicitization. Our contribution is twofold. On one hand we tackle the aforementioned issue in the case of 2D curves and 3D surfaces by Minkowski decomposing Q, thus detecting the Minkowski summand relevant to implicitization: we design and implement in Sage a new, public domain, practical, potentially generalizable and worst-case optimal algorithm for Minkowski decomposition in 3D based on integer linear programming. On the other hand, we formulate basic geometric predicates, namely membership and sidedness for given query points, as rank computations on the interpolation matrix, thus avoiding to expand the implicit polynomial. This approach is implemented in Maple.