geoTangle : Interactive Design of Geodesic Tangle Patterns on Surfaces

Tangles are complex patterns, which are often used to decorate the surface of real-world artisanal objects. They consist of arrangements of simple shapes organized into nested hierarchies, obtained by recursively splitting regions to add progressively finer details. In this article, we show that 3D digital shapes can be decorated with tangles by working interactively in the intrinsic metric of the surface. Our tangles are generated by the recursive application of only four operators, which are derived from tracing the isolines or the integral curves of geodesics fields generated from selected seeds on the surface. Based on this formulation, we present an interactive application that lets designers model complex recursive patterns directly on the object surface without relying on parametrization. We reach interactive speed on meshes of a few million triangles by relying on an efficient approximate graph-based geodesic solver.

Evolutionary Sequence of Spacetime and Intrinsic Spacetime and Associated Sequence of Geometries in Metric Force Fields I

Two classes of three-dimensional metric spaces are identified. They are the conventional three-dimensional metric space and a new ‘three-dimensional’ absolute intrinsic metric space. Whereas an initial flat conventional proper metric space IE′3 can transform into a curved three-dimensionalRiemannian metric space IM′3 without any of its dimension spanning the time dimension (or in the absence of the time dimension), in conventional Riemann geometry, an initial flat ‘three-dimensional’ absolute intrinsic metric space ∅IˆE3 (as a flat hyper-surface) along the horizontal, evolves into a curved ‘three-dimensional’ absolute intrinsic metric space ∅IˆM3, which is curved (as a curved hyper-surface) toward the absolute intrinsic metric time ‘dimension’ along the vertical, and it is identified as ‘three-dimensional’ absolute intrinsic Riemannian metric space. It invariantly projects a flat ‘three-dimensional’ absolute proper intrinsic metric space ∅IE′3ab along the horizontal, which is made manifested outwardly in flat ‘three-dimensional’ absolute proper metric space IE′3ab, overlying it, both as flat hyper-surfaces along the horizontal. The flat conventional three-dimensional relative proper metric space IE′3 and its underlying flat three-dimensional relative proper intrinsic metric space ∅IE′3 remain unchanged. The observers are located in IE′3. The projective ∅IE′3ab is imperceptibly embedded in ∅IE′3 and IE′3ab in IE′3. The corresponding absolute intrinsic metric time ‘dimension’ is not curved from its vertical position simultaneously with ‘three-dimensional’ absolute intrinsic metric space. The development of absolute intrinsic Riemannian geometry is commenced and the conclusion that the resulting geometry is more all-encompassing then the conventional Riemannian geometry on curved conventional metric space IM′3 only is reached.

Characterizations of canonically compactifiable graphs via intrinsic metrics and algebraic properties

We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any finite measure intrinsic metric. Furthermore, we show that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra. These results answer questions in a recent work of Georgakopoulos, Haeseler, Keller, Lenz, and Wojciechowski.

REALIZING METRICS OF CURVATURE ON CLOSED SURFACES IN FUCHSIAN ANTI-DE SITTER MANIFOLDS

We prove that any metric with curvature less than or equal to $-1$ (in the sense of A. D. Alexandrov) on a closed surface of genus greater than $1$ is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension $(2+1)$ with sectional curvature $-1$ . The proof is by approximation, using a result about isometric immersion of smooth metrics by Labourie and Schlenker.

Ancient Caloric Functions on Graphs With Unbounded Laplacians

We study ancient solutions of polynomial growth to both continuous-time and discrete-time heat equations on graphs with unbounded Laplacians. We extend Colding and Minicozzi’s theorem [12] on manifolds and the result [22] on graphs with normalized Laplacians to the setting of graphs with unbounded Laplacians: for a graph admitting an intrinsic metric, which has polynomial volume growth, the dimension of the space of ancient solutions of polynomial growth is bounded by the dimension of harmonic functions with the same growth up to some factor.