The topological correctness of PL approximations of isomanifolds

Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f : Rd → Rd−n. A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation T of the ambient space Rd. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary

Detecting Weakly Simple Polygons

A closed curve in the plane is weakly simple if it is the limit (in the Fr\'echet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201

LIPIcs

Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. submanifolds of ℝ^d defined as the zero set of some multivariate multivalued smooth function f: ℝ^d → ℝ^{d-n}, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M is to consider its Piecewise-Linear (PL) approximation M̂ based on a triangulation of the ambient space ℝ^d. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ = 1/D (and unavoidably exponential in n). Since it is known that for δ = Ω (d^{2.5}), M̂ is O(D²)-close and isotopic to M, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M̂ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art

Abstract Morphing Using the Hausdorff Distance and Voronoi Diagrams

This paper introduces two new abstract morphs for two 2-dimensional shapes. The intermediate shapes gradually reduce the Hausdorff distance to the goal shape and increase the Hausdorff distance to the initial shape. The morphs are conceptually simple and apply to shapes with multiple components and/or holes. We prove some basic properties relating to continuity, containment, and area. Then we give an experimental analysis that includes the two new morphs and a recently introduced abstract morph that is also based on the Hausdorff distance [Van Kreveld et al., 2022]. We show results on the area and perimeter development throughout the morph, and also the number of components and holes. A visual comparison shows that one of the new morphs appears most attractive

Tristan da Cunha hotspot : Mantle plume or shallow plate tectonics?

Tristan da Cunha is a small volcanic island in the South Atlantic Ocean close to the Mid- Atlantic Ridge. It is part of an area, which is characterized by widely scattered seamounts and small islands at the western end of the Walvis Ridge - Tristan/Gough hotspot track. Tristan da Cunha represents the end member of a classical hotspot track with an underlying plume: The active volcanic island Tristan da Cunha at the youngest end of the track is linked to the Cretaceous Etendeka flood basalt province in northwestern Namibia at its oldest end. But the genesis of the island itself has so far been puzzling. It is hotly debated if the island sits actually above a deep-seated mantle plume or if it is caused by shallow plate tectonics. To understand the Tristan da Cunha hotspot, a multi-disciplinary geophysical study has been conducted in 2012 and 2013 on board the German research vessel Maria S. Merian to acquire marine magnetotelluric and seismological data. The aim was to reveal the upper mantle structure with electrical density and velocity perturbations. Within this study I focused on the seismological dataset. At first, I performed a P-wave finite-frequency tomography with cross-correlated travel time residuals of teleseismic earthquakes. This allows to resolve the upper mantle structure beneath the island in terms of velocity perturbations and clarifies the existence of a mantle plume. I also investigated the local seismicity in the Tristan region to identify tectono-magmatic processes at the Mid-Atlantic Ridge and Close to the islands and seamounts. Moreover, I compared and combined my tomographic results with electromagnetic results to identify zones of partial melt and to understand plume processes in the upper mantle beneath the Tristan da Cunha hotspot. The tomographic results provide evidence for the existence of the Tristan conduit southwest of the archipelago. Its shape is cylindrical with a radius ca. 100 km down to a depth of 250 km. The structure ramifies in narrow veins below that depth. A recent link from the conduit to a seamount chain shows, that melt is channelled towards seamounts and islands in the study area. High seismicity within an oceanic plate segment north of Tristan da Cunha can be related to the internal stresses of the fragment. Differently directed forces act at the northern boundary of this plate. An earthquake free zone coincides spatially with the location of the Tristan mantle plume. This indicates a ductile regime in the lithosphere above the plume. Furthermore, hints for an incipient ridge jump towards a parallel line to the actual location of the Tristan plume were found. Several earthquakes were localised close to the archipelago of Tristan da Cunha. The locations of these earthquakes are related to young surface eruptions like small volcanic cones or seamounts