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Living materials such as membranes, cytoskeletal assemblies, cell collectives and tissues can often be described as active solids -- materials that are energized from within, with elastic response about a well defined reference…
In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside $\mathbb{R} \times \mathbb{R}^3 \equiv \mathbb{R}^4$, which forms a triple. We want to define an ambient isotopic…
Information transfer between triangle meshes is of great importance in computer graphics and geometry processing. To facilitate this process, a smooth and accurate map is typically required between the two meshes. While such maps can…
In the field of spatial computing, one of the most essential tasks is the pose estimation of 3D objects. While rigid transformations of arbitrary 3D objects are relatively hard to detect due to varying environment introducing factors like…
Mathematically representing the shape of an object is a key ingredient for solving inverse rendering problems. Explicit representations like meshes are efficient to render in a differentiable fashion but have difficulties handling topology…
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve has attracted much interest. In the present paper, we propose a new method to construct a surface…
There is no settled universal 3D representation for geometry with many alternatives such as point clouds, meshes, implicit functions, and voxels to name a few. In this work, we present a new, compelling alternative for representing shapes…
This paper introduces the new interactive Java sketching software KamiWaAi, recently developed at the University of Fukui. Its graphical user interface enables the user without any knowledge of both mathematics or computer science, to do…
We consider the popular and classical method of alternating projections for finding a point in the intersection of two closed sets. By situating the algorithm in a metric space, equipped only with well-behaved geodesics and angles (in the…
The square-peg problem asks if every Jordan curve in the plane has four points which are the vertices of a square. The problem is open for continuous Jordan curves, but it has been resolved for various regularity classes of curves between…
The square root velocity transformation provides a convenient and numerically efficient approach to functional and shape data analysis of curves. We study fundamental geometric properties of curves under this transformation. Moreover,…
This paper presents an alternative approach to simplify the proofs of some important results related to polynomial mappings in Computational Algebraic Geometry such as Polynomial Implicitization, Image Closure and some properties of the…
Utilizing recently developed abstract notions of sectional curvature, we introduce a method for constructing a curvature-based geometric profile of discrete metric spaces. The curvature concept that we use here captures the metric relations…
Geometric frameworks for analyzing curves are common in applications as they focus on invariant features and provide visually satisfying solutions to standard problems such as computing invariant distances, averaging curves, or registering…
The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to…
Quantum Graphity is an approach to quantum gravity based on a background independent formulation of condensed matter systems on graphs. We summarize recent results obtained on the notion of emergent geometry from the point of view of a…
Object data analysis is concerned with statistical methodology for datasets whose elements reside in an arbitrary, unspecified metric space. In this work we propose the object shape, a novel measure of shape/symmetry for object data. The…
In this work, an evolutionary art project is presented where images are approximated by transparent, overlapping and geometric shapes of different types, e.g., polygons, circles, lines. Genotypes representing features and order of the…
Let $C$ be a smooth, convex curve on either the sphere $\mathbb{S}^{2}$, the hyperbolic plane $\mathbb{H}^{2}$ or the Euclidean plane $\mathbb{E}^{2}$, with the following property: there exists $\alpha$, and parameterizations $x(t), y(t)$…
The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible…