Related papers: Hands-on Tropical Geometry
We study the tropical lines contained in smooth tropical surfaces in R^3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however,…
3D printing of surfaces has become an established method for prototyping and visualisation. However, surfaces often contain certain degenerations, such as self-intersecting faces or non-manifold parts, which pose problems in obtaining a 3D…
The paper develops a method for the numerical simulation of a free-surface flow of incompressible viscous fluid around a streamlined body. The body is a rigid stationary construction partially submerged in the fluid. The application we are…
We tackle the challenging problem of creating full and accurate three dimensional reconstructions of botanical trees with the topological and geometric accuracy required for subsequent physical simulation, e.g. in response to wind forces.…
Three-dimensional (3D) printing has emerged as a powerful tool for material, food, and life science research and development, where the technology democratization necessitates the advancement of open-source platforms. Herein, we developed a…
We propose a method for constructing high-quality, closed-surface meshes from confined 3D point clouds via a physically-based simulation of flexible foils under spatial constraints. The approach integrates dynamic elasticity,…
This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence…
The simulation and parameter optimization of coupled ocean circulation and ecosystem models in three space dimensions is one of the most challenging tasks in numerical climate research. Here we present a scientific toolkit that aims at…
We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental…
Direct manipulation has been established as the main interaction paradigm for Computer-Aided Design (CAD) for decades. It provides fast, incremental, and reversible actions that allow for an iterative process on a visual representation of…
By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.
Three-dimensional (3D) printing is a powerful development tool both in industry, as well as in biomedical research. Additive-lathe 3D printing is an emerging sub-class of 3D printing whereby material is layered outward from the surface of a…
The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is…
This paper surveys {\it tropical modifications}, which have already become a folklore in tropical geometry. Tropical modifications are used in tropical intersection theory, tropical Hodge theory, and in the study of singularities. They…
Mechanical metamaterials are structures designed to exhibit an exotic response, such as topological soft modes at a surface. Here we explore single-material 3D prints of these topological structures by translating a ball-and-spring model…
We provide explicit faithful re-embeddings for all hyperelliptic curves of genus at most three and an algorithmic way to construct them. Both in the faithful tropicalization algorithm and the proofs of correctness, we showcase OSCAR-methods…
We describe an algorithm that takes as input an open book decomposition of a closed oriented 4-manifold and outputs an explicit trisection diagram of that 4-manifold. Moreover, a slight variation of this algorithm also works for open books…
The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections…
Mass production of augmented reality (AR) waveguides has been challenging due to the intricate nature of the fabrication technique and the high precision required for its optical characteristics. In this paper, we have presented a novel and…
We present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical…