Geodesic Circles Research Articles

Persistent Homology with Selective Rips Complexes Detects Geodesic Circles

This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^1$$\\end{document}) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space using persistent homology. Under fairly mild conditions, we show that such a loop either terminates a 1-dimensional homology class or gives rise to a 2-dimensional homology class in persistent homology. The main tool in this detection technique are selective Rips complexes, new custom made complexes that function as an appropriate combinatorial lens for persistent homology to detect the above mentioned loops. The main argument is based on a new concept of a local winding number, which turns out to be an invariant of certain homology classes.

Volume preserving flows for convex curves and surfaces in the hyperbolic space

In this paper, we study the curvature flows of convex curves and surfaces in the hyperbolic space. In the first part, we consider the area preserving and length preserving κα-type curvature flows of smooth closed convex curves in the hyperbolic plane H2 and show that these two types of flows evolve convex curve to a geodesic circle exponentially in C∞ topology. In the second part, we study the volume preserving Gauss curvature flow of smooth closed convex surfaces in H3 and show that the solution remains convex and converges to a geodesic sphere exponentially in C∞ topology.

Footprints of Geodesics in Persistent Homology

Given a metric space X and a subspace $$A\subset X$$ , we prove that A can generate various algebraic elements in persistent homology of X. We call such elements (algebraic) footprints of A. Our results imply that footprints typically appear in dimensions above $$\dim (A)$$ . Higher dimensional persistent homology thus encodes lower dimensional geometric features of X. We pay special attention to a specific type of geodesics in a geodesic surface X called geodesic circles. We explain how they may generate non-trivial odd-dimensional and two-dimensional footprints. In particular, we can detect even some contractible geodesics using two- and three-dimensional persistent homology. This provides a link between persistent homology and the length spectrum in Riemannian geometry.

Topography measurement of helical grooves on a hemisphere based on stitching interference microscopy

Hemispherical bearings fabricated with helical grooves on the working surface show improved aerodynamic performance which depends to a large extent on the machining accuracy of the grooves. Topography measurement of the entire groove from macro to micro scale is thus important to evaluate the bearing performance. White light interference microscopy is widely used in topography measurement. However, subaperture stitching is necessary to extend the field of view (FOV) for large area measurement but limited to planar surfaces. This paper presents a method of interference microscopy combined with subaperture stitching. The long groove is divided into a series of subapertures with overlapping regions. Each subaperture meets the requirement of FOV and measurable slope limit of the microscope. We show that the helical curve is approximately located on a geodesic circle of the hemisphere. Accordingly a special fixture is designed with a rotary table to rotate the groove around the axis perpendicular to the geodesic circle. A four-axis scanning platform is integrated with the interference microscope. Based on the principle of minimal deviations in overlapping regions, six-degree-of-freedom motion error introduced in subaperture scanning is simultaneously corrected through an iterative stitching algorithm. It is experimentally verified by measuring a real part with helical grooves etched on a hemisphere bearing rotor. Complete high-resolution topography of the entire groove is reconstructed and the stitching map shows no overlap sliding effect, which is instructive for assessment of the machining error and further evaluation of the bearing performance.

On an inverse curvature flow in two-dimensional space forms

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetric inequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $$n=2$$ case of a conjecture of Girão–Pinheiro.

Curves of geodesic centers and Poncelet ellipses

In this paper, we consider smooth, closed, strictly convex curves contained inside the unit circle T and their associated curves of geodesic centers (or dual curves). We obtain a formula for the boundary and envelope of the union of the region bounded by the geodesic circles associated with our curves. This description includes a formula for the boundary of the hyperbolic convex hull of points identified by finite Blaschke products. We describe a setting and conditions under which one can produce an infinite chain of ellipses such that each ellipse is inscribed in a convex polygon that is itself inscribed in another polygon (a chain of Poncelet ellipses). We apply these results to operators that are compressions of the shift operator on model spaces and describe necessary and sufficient conditions for the numerical range to contain 0.

Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature

Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions e λ e_{\lambda} over a closed smooth curve 𝛾 which satisfies a natural curvature condition, go to 0 at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ → ∞ \lambda\to\infty if 0 < | ν | λ < 1 - δ 0<\frac{\lvert\nu\rvert}{\lambda}<1-\delta for any fixed 0 < δ < 1 0<\delta<1 . Our result implies, for instance, that the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) .

Ultra-wide band gap in a two-dimensional phononic crystal with hexagonal lattices

A new two-dimensional (2D) phononic crystal (PC) constituted by the periodic hexagonal lattices, which is inspired by the scissor-like geodesic circle of the Hoberman sphere, is proposed in this study. The band structures, transmission spectrum as well as displacement field of the proposed PC are analyzed by employing the finite element method (FEM) with the Bloch theorem. Furthermore, the formation mechanism of the band gaps (BGs) is investigated according to the eigenmodes at the BG edges analysis. Results show that the first BG of the proposed PC can be achieved in the range of 244.6 Hz–1490.0 Hz with a gap-mid gap ratio (BG%) of 143.6 %, and it is produced via the rigid body resonance. Also, a BG span from 244.6 Hz to 2543.9 Hz with a BG% of 164.9 % is obtained when an approximate flat band between the first and second BGs is neglected, showing an ultra-wide BG in the low-mid frequency range. We further show that the position and width of the BGs can be adjusted within a broad frequency range by altering the key geometric parameters.

Sketching Manipulators for Localized Blendshape Editing

We introduce a method that localizes the direct manipulation of blendshape models for facial animation with a customized sketch-based interface. Direct manipulation methods address the problem of cumbersome weight editing process using the traditional tools with a practical “pin-and-drag” operation directly on the 3D facial models. However, most of the direct manipulation methods have a global deformation impact, which lead to unintuitive and unexpected results. To this end, we propose a new way to localize the theory of direct manipulation method using geodesic circles for confining the edits to the local geometry. Inspired by artists’ brush painting on canvas, we additionally introduce a sketch-based interface as an application that provides direct manipulation and produces expressive facial poses efficiently and intuitively. Our method allows the artists to simply sketch directly onto the 3D facial model and automatically produces the expeditious manipulation until the desired facial pose is obtained. We show that localized blendshape direct manipulation has the potential to reduce the time-consuming blendshape editing process to an easy freehand stroke drawing. Finally, we present a wide range of facial poses on various models that are created rapidly using our method.

Geodesics and geodesic circles in a geodesically convex surface: a sub-mixing property

Geodesics and geodesic circles in a geodesically convex surface: a sub-mixing property

On Concircular Transformations in Finsler Geometry

A geodesic circle in Finsler geometry is a natural extension of that in a Euclidean space. In this paper, we study geodesic circles and (infinitesimal) concircular transformations on a Finsler manifold. We characterize a concircular vector field with some PDEs on the tangent bundle, and then we obtain respectively necessary and sufficient conditions for a concircular vector field to be conformal and a conformal vector field to be concircular. We also show conditions for two conformally related Finsler metrics to be concircular, and obtain some invariant curvature properties under conformal and concircular transformations.

An approximation of geodesic circle passing through three points on an ellipsoid

Abstract In this paper we present an approximation method for a geodesic circle passing through three points on an oblate ellipsoid. Our method uses a prolate ellipsoid passing through the three points, and the new approximation curve is the intersection of the oblate and prolate ellipsoids, which can be obtained algebraically without iterations. The advantage of our approximation method is that it yields a significantly smaller approximation error. Compared to the plane section curve passing through the three points on the oblate ellipse, our method reduces the approximation error by at least 98 % 98\hspace{0.1667em}\% when the radii of geodesic circles are 100 km∼1000 km on the surface of the Earth. We illustrate the results using numerical examples.

Multidimensional scaling on metric measure spaces

Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its optimality properties and goodness of fit. Further, we present a notion of MDS on infinite metric measure spaces that generalizes these optimality properties. As a consequence we can study the MDS embeddings of the geodesic circle $S^1$ into $\mathbb{R}^m$ for all $m$, and ask questions about the MDS embeddings of the geodesic $n$-spheres $S^n$ into $\mathbb{R}^m$. Finally, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space $X$, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of $X$?

The asymptotic behavior of geodesic circles in any 2-torus: a sub-mixing property

The asymptotic behavior of geodesic circles in any 2-torus: a sub-mixing property

On the n-body problem on surfaces of revolution

We explore the n-body problem, n≥3, on a surface of revolution with a general interaction depending on the pairwise geodesic distance. Using the geometric methods of classical mechanics we determine a large set of properties. In particular, we show that Saari's conjecture fails on surfaces of revolution admitting a geodesic circle. We define homographic motions and, using the discrete symmetries, prove that when the masses are equal, they form an invariant manifold. On this manifold the dynamics are reducible to a one-degree of freedom system. We also find that for attractive interactions, regular n-gon shaped relative equilibria with trajectories located on geodesic circles typically experience a pitchfork bifurcation. Some applications are included.

AdS 3-manifolds and Higgs bundles

In this paper we investigate the relationships between closed AdS 3 3 -manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for their volume. We give natural foliations of the AdS structure with time-like geodesic circles and we use these circles to construct equivariant minimal immersions of the Poincaré disc into the Grassmannian of time-like 2-planes of R 2 , 2 \mathbb {R}^{2,2} .

On different topological classes of spherical geodesic paths and circles in [formula omitted

A discrete spherical geodesic path (DSGP) between two voxels s and t lying on a discrete sphere is a/the shortest path from s to t, comprising voxels of the discrete sphere intersected by the discrete geodesic plane passing through s, t, and the center of the sphere. We consider two classes of discretization, namely naive and standard, for both the sphere and the geodesic plane, which gives rise to four distinct topological classes of DSGP. We show that the naive–naive class does not guarantee the existence of a DSGP, whereas the other three classes do. We derive the upper bounds of the distance of a DSGP belonging to each class, from the real sphere and the real plane, for different neighborhood conditions. We propose an efficient integer-based algorithm to compute the DSGP for any class-and-neighborhood combination. Novel number-theoretic characterization of discrete sphere has been used for searching the voxels comprising a DSGP. The algorithm is output-sensitive, having its time and space complexities both linear in the length of the DSGP. It can also be extended for constructing discrete 3D circles of arbitrary orientations, specified by a few appropriate input parameters. Experimental results and related analysis demonstrate its efficiency and versatility.

On the construction of minimal surfaces from geodesics

In a recent article (2013), Li et al. [1] approximate minimal surfaces from geodesic boundaries, with applications to garment design in mind. We go over this work and existing methods for constructing minimal surfaces from geodesics. First, we justify why minimal surfaces and the problem of finding the surface with minimal area (i.e., solving Plateau's problem) have little to do with garment design. Second, we recall that Plateau's problem makes little sense for boundaries such as those considered in [1], composed of unclosed curves of finite length or disconnected pieces of them (with no other positional restriction). Finally, we note that the construction of a minimal surface (with zero mean curvature) from a prescribed geodesic is a particular instance of a classical problem in differential geometry, already solved by Björling. In particular, for a geodesic circle or helix the resulting minimal surfaces are well-known (catenoid and helicatenoid, respectively), so no approximations are required.

A classification of complete Finsler manifolds through the conformal theory of curves

In the first part of this work we are going to touch the conformal theory of curves on Finsler geometry, emphasizing on the notion of circles preserving transformations, recently studied by the present author and Z. Shen. Next, the conformal changes of metrics which leave invariant geodesic circles known as concircular transformation are characterized by a second order differential equation. As an application a classification of complete Finsler manifolds admitting such transformations is obtained.

Sharp bounds for the intersection of nodal lines with certain curves

Let Y be a hyperbolic surface and let \phi be a Laplacian eigenfunction having eigenvalue -1/4-\tau^2 with \tau>0 . Let N(\phi) be the set of nodal lines of \phi . For a fixed analytic curve \gamma of finite length, we study the number of intersections between N(\phi) and \gamma in terms of \tau . When Y is compact and \gamma a geodesic circle, or when Y has finite volume and \gamma is a closed horocycle, we prove that \gamma is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between N(\phi) and \gamma is O(\tau) . This bound is sharp.