The article introduces a new shape signature method using homology nerves and proximal relators.
problem Describing the shape of finite, bounded planar objects.
method Planar shape signatures derived from homology nerves with proximal relators.
result Every finite, bounded planar shape has a signature derived from the homology group.
Deep model predicts shapes of curves with multiple covariates.
problem Predicting shapes of planar curves with various covariates.
method Deep learning model using complex-valued functions, conditional covariance smoother with modality-specific encoders.
result Model accurately predicts shapes of curves with multimodal covariates.
We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point. Our theorem is the analog of the result of Grayson for curvature flow of clo…
New method uses contours of segmented images for X-ray classification.
problem Classifying X-ray images of segmented radiography.
method Develops a new approach for image analysis of multivariate planar curves, addressing alignment issues.
result Demonstrates the robustness and appeal of the proposed method through detection of cardiomegaly and numerical experiments.
New kernel method for shape classification on Kendall shape space.
problem Classification of shapes on non-Euclidean Kendall shape space.
method Extrinsic Veronese Whitney Gaussian kernel for KRRC on Σ2k. result KRRC classifier performs well on real Kendall shape data.
The paper examines singular points in Wigner caustics and affine equidistants of planar curves.
problem Analyzing singular points in Wigner caustics and affine equidistants of planar curves.
method Generalizing the Blaschke-Süss theorem to study convex curves and their antipodal pairs.
result Existence of antipodal pairs in convex curves is generalized.
Reduces 3-body problem to shape and spherical geometry.
problem Investigates the motion of three bodies in a plane.
method Geometric reduction using equivariant Riemannian geometry.
result Time parametrization of moduli curve determined by shape curve and potential function.
Study totally umbilic submanifolds using planar pseudo-geodesics.
problem Characterize totally umbilic isometric immersions with parallel normalized mean curvature vector.
method Introduce planar pseudo-geodesics and analyze their properties; prove the equivalence of totally umbilic immersions and planar geodesic extrinsic shapes.
result An isometric immersion is totally umbilic if and only if every geodesic of the manifold has planar extrinsic shape.
We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution …
Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint - a computable approximation to a conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22 for a few examples). In this paper, we make the first significant step in this pr…
We create a smooth manifold of triangular meshes with a geodesically complete metric.
problem Representing and manipulating 2D shapes as triangular meshes.
method Developed a geodesically complete Riemannian metric for triangular meshes.
result The metric preserves mesh connectivity and avoids mesh degradation.
Paper introduces length measures for curves and convex shapes, proving isoperimetric and distance properties.
problem Characterizing and comparing convex shapes using length measures.
method Developed length measures for curves and convex shapes, derived properties, and introduced a new distance metric.
result Unique convex curve maximizes signed area among curves with same length measure.
A new method for analyzing shapes and forms using additive models on manifolds.
problem Analyzing shapes and forms under geometric transformations.
method Extending generalized additive regression to models for shapes/forms using squared geodesic distance and Riemannian L2-Boosting algorithm. result Automated model selection and intuitive visualization of covariate effects in shape/form space.
Infinitely many 3D shapes have multiple ways to be filled with special surfaces.
problem Understanding how many ways 3D shapes can be filled with special surfaces.
method Examined 3D shapes supported by planar open books and found multiple ways to fill them with special surfaces.
result Found infinitely many 3D shapes that can be filled with multiple, non-homeomorphic special surfaces.
Unified description of aesthetic curves through self-affinities.
problem Characterizing log-aesthetic curves and their properties.
method Reformulating and proving self-affinities of planar curves, integrating equiaffine geometry.
result Unified characterization of constant curvature curves in similarity and equiaffine geometries.
By using two different invariants for the Rubik's Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing …
This article presents certain recent methodologies and some new results for the statistical analysis of probability distributions on manifolds. An important example considered in some detail here is the 2-D shape space of k-ads, comprising all configurations of k planar landmarks (k>2)-modulo translation, scaling a…
Any subset of the plane can be approximated by a set of square pixels. This transition from a shape to its pixelation is rather brutal since it destroys geometric and topological information about the shape. Using a technique inspired by Morse Theory, we algorithmically produce a PL approximation of the original shape …
The paper analyzes the emergence of almost-honeycomb structures in low-energy planar clusters.
problem Understanding the formation of shapes resembling honeycombs in low-energy configurations.
method Detailed quantitative estimates and a revision of the global isoperimetric principle for honeycomb clusters.
result The majority of chambers in low-energy planar clusters are generalized hexagons, closely resembling regular hexagons.
The paper constructs λ-hypersurfaces for λ>0 and λ<0.
problem Exploring λ-hypersurfaces in different λ-values and their properties. method Constructing complete embedded and non-convex λ-hypersurfaces diffeomorphic to a cylinder and doughnut-shaped. result For λ>0, complete embedded and non-convex λ-hypersurfaces are constructed, diffeomorphic to a cylinder. New method for sensing non-planar surfaces using ERT.
problem Limited computational techniques for planar surfaces in ERT-based sensing skins.
method Generalized ERT to non-planar surfaces using Riemannian geometry.
result Feasibility and applicability of ERT-based sensing skins for non-planar geometries demonstrated.
Planar neural networks learn image transformations from sequences.
problem Learning image transformations for mental simulation.
method Using planar neural networks, the study investigates various factors affecting the learning of image transformations.
result The approach can effectively learn and transfer image transformations, including translation, rotation, and scaling.
Evolution of planar curves under a nonlocal geometric equation is investigated. It models the simultaneous contraction and growth of carbonate particles called ooids in geosciences. Using classical ODE results and a bijective mapping we demonstrate that the steady parameters associated with the physical environment det…
The paper introduces vortex nerve complexes and new Betti numbers in CW spaces.
problem Understanding the structure and properties of CW complexes and their nerves.
method Introducing vortex nerve complexes and defining new Betti numbers for CW complexes.
result New Betti numbers (vortex Bvtex, vortex nerve BvNrv, shape Bsh) are introduced and studied. Study the hanging chain shape around a circle.
problem Finding the shape of a curve extremizing potential energy to a circle.
method Analyzes curves minimizing potential energy to a circle, considering both inside and outside.
result Describes shapes of curves for different powers of distance to the circle.
We describe an algorithm that associates to each positive real number r and each finite collection Cr of planar pixels of size r a planar piecewise linear set Sr with the following additional property: if Cr is the collection of pixels of size r that touch a given compact semialgebraic set S, then the …
Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher dimensional curves, surfaces, activities, character motions and many other objects. In …
This paper uses Ghrist barcodes to track persistent shapes in video frames.
problem Detecting and tracking persistent shapes in video frames.
method Introduces Ghrist barcodes for persistent Betti numbers derived from vortex nerve complexes in triangulated video frames.
result Persistent Betti numbers of vortex nerves are k+2 for k edges. Study the limiting shape of solutions to the L_p-Minkowski problem as p approaches negative infinity.
problem Understanding the limiting shape of solutions to the L_p-Minkowski problem as p → -∞.
method Group-invariant method to study the asymptotic shape of solutions.
result Existence of a solution Ω^(p) to the L_p-Minkowski problem that converges to a regular polytope T as p → -∞.
This paper defines ribbons and ribbon complexes in CW spaces and analyzes their topological properties.
problem Characterizing and analyzing topological structures in CW spaces.
method Introducing planar ribbons, ribbon complexes, and ribbon nerves in Alexandroff-Hopf-Whitehead CW spaces, and studying their topological properties.
result Characterization of ribbons and ribbon nerves by Betti numbers and homotopy types.
We introduce elastic geodesic grids for easy-to-fabricate, deployable structures.
problem Approximating freeform surfaces with deployable structures.
method Geodesic curves on target surfaces, kinematic mechanism, differential geometry.
result Elastic geodesic grids can approximate freeform surfaces easily and deployably.
Kirigami-inspired math reveals shortest paths and ultimate shapes of cut paper.
problem Geodesics and isometric immersions in paper with cuts.
method Constructive proof of geodesics and rectification of polygonal geodesics.
result Polygonal geodesics can be rectified into a straight line by flat-folding.
Gradient flow expands curves to round shapes.
problem Expanding curves to round shapes.
method Steepest descent L2-gradient flow of entropy.
result Flow converges to a round expanding circle for various initial curves.
Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.
problem Eigenvalue continuity and shape optimization for Laplace and Steklov problems.
method Variational eigenvalue analysis, Sobolev space convergence, shape optimization techniques.
result Sharp isoperimetric inequalities for Steklov eigenvalues, upper bound 8πk for k-th perimeter-normalized eigenvalue. Extends shape analysis to framed space curves using quaternionic arithmetic.
problem Matching and classifying shapes of framed space curves.
method Extends square root transform to framed curves using quaternionic arithmetic and Hopf fibration properties. Describes geodesics in framed curve space explicitly.
result Explicit descriptions of geodesics in framed curve space and averages of collections of curves.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width, and is conjectured to have as its solution a complicated shape derived by Gerver in 1992. We extend Gerver's techniques by deriving a family of six differe…
Efficient method for shape modeling invariant to rigid motion.
problem Statistical shape modeling for rigidly moving shapes.
method Non-Euclidean Lie group analysis of metric distortion and curvature.
result Outperforms state-of-the-art classifiers in sparse data.
We give a number of examples of isospectral pairs of plane domains, and a particularly simple method of proving isospectrality. One of our examples is a pair of domains that are not only isospectral but homophonic: Each domain has a distinguished point such that corresponding normalized Dirichlet eigenfunctions take eq…
Optimal thresholds ensure curves remain embedded in flows.
problem Preserving the embeddedness of elastic flows of curves.
method Variational characterization and minimization of bending energy.
result Optimal thresholds for preserving embeddedness are found.
Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.
problem Minimizing elastic bending energy for open planar curves with obstacles.
method Investigation of global minimizers and explicit solutions for different values of the penalization parameter.
result Explicit threshold for λ above which minimizers touch the obstacle, regardless of obstacle shape. Can certain shapes be drawn with a pencil and eraser?
problem Characterizing which planar sets can be drawn with a pencil and eraser.
method Analyzes the properties of sets drawable with a pencil and eraser, using open and closed unit disks.
result Drawability cannot be characterized by local obstructions.
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an L2-gradient flow for Euler's elastic energy. result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
Second order Sobolev metrics on the space of regular unparametrized planar curves have several desirable completeness properties not present in lower order metrics, but numerics are still largely missing. In this paper, we present algorithms to numerically solve the initial and boundary value problems for geodesics. Th…
New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.
problem Designing aesthetic shapes in equiaffine geometry.
method Introducing a new symmetry (ESA) to characterize planar curves.
result The new class of curves includes the quadratic curve and logarithmic spiral.
Ancient convex solutions to flow equations are limited to simple shapes.
problem Characterizing ancient convex solutions to flow equations.
method Analyzing mean curvature flow and curvature functions of convex hypersurfaces.
result Ancient convex solutions to flow equations are limited to spherical, cylindrical, or planar shapes.
We study metrics on shape space of immersions that have a particularly simple horizontal bundle. More specifically, we consider reparametrization invariant Sobolev metrics G on the space Imm(M,N) of immersions of a compact manifold M in a Riemannian manifold (N,g). The tangent space $T…
Computes elastic grids that approximate 3D surfaces without physical simulations.
problem Creating planar grids that fit complex 3D surfaces efficiently.
method Uses differential geometry to minimize bending energy and nestle to the surface.
result Elastic grids can approximate 3D surfaces without physical simulations.