Geodesic Polar Coordinates Research Articles

Hadamard integrator for time-dependent wave equations: Lagrangian formulation via ray tracing

Starting from the time-domain Kirchhoff-Huygens representation of wave solutions, we propose a novel Hadamard integrator for the self-adjoint time-dependent wave equation in an inhomogeneous medium. First, we create a new asymptotic series based on the Gelfand-Shilov function, dubbed Hadamard's ansatz, to approximate the Green's function of the time-dependent wave equation. Accordingly, the governing equations and related initializations for the eikonal and Hadamard coefficients are derived using the properties of the Gelfand-Shilov generalized function. Second, incorporating the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation, we develop an original Hadamard integrator for the Cauchy problem of the time-dependent wave equation and derive the corresponding Lagrangian formulation in geodesic polar coordinates. Third, to construct the Hadamard integrator in the Lagrangian formulation efficiently, we use a short-time ray tracing method to obtain equal-time wavefront locations accurately, and we further develop fast algorithms to compute Chebyshev-polynomial based low-rank representations of both wavefront locations and variants of Hadamard coefficients. Fourth, equipped with these low-rank representations, we apply the Hadamard integrator to efficiently solve time-dependent wave equations with highly oscillatory initial conditions, where the time step size is independent of the initial conditions. By judiciously choosing the medium-dependent time step, our new Hadamard integrator can propagate wave field beyond caustics implicitly and advance spatially overturning waves in time naturally. Moreover, since the integrator is independent of initial conditions, the Hadamard integrator can be applied to many different initial conditions once it is constructed. Both two-dimensional and three-dimensional numerical examples illustrate the accuracy and performance of the proposed method.

Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by where q i and p i are generic canonical variables, γ n are arbitrary coefficients, and . For N = 2, being both different from zero, this reduces to the classical Zernike system. We prove that always provides a superintegrable system (for any value of γ n and N) by obtaining the corresponding constants of the motion explicitly, which turn out to be of higher-order in the momenta. Such generic results are not only applied to the Euclidean plane, but also to the sphere and the hyperbolic plane. In the latter curved spaces, is expressed in geodesic polar coordinates showing that such a new superintegrable Hamiltonian can be regarded as a superposition of the isotropic 1 : 1 curved (Higgs) oscillator with even-order anharmonic curved oscillators plus another superposition of higher-order momentum-dependent potentials. Furthermore, the symmetry algebra determined by the constants of the motion is also studied, giving rise to a th-order polynomial algebra. As a byproduct, the Hamiltonian is interpreted as a family of superintegrable perturbations of the classical Zernike system. Finally, it is shown that (and so the Zernike system as well) is endowed with a Poisson -coalgebra symmetry which would allow for further possible generalizations that are also discussed.

More on Superintegrable Models on Spaces of Constant Curvature

A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalised Kepler potential. The angular components, on the contrary, are given implicitly by a transcendental, in general, equation. In the present note, devoted to the previously less studied models with the radial potential of the generalised Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of elementary functions. For an appropriate choice of parameters, the family reduces to an asymmetric spherical Higgs oscillator.

Superintegrability on the Dunkl oscillator model in three-dimensional spaces of constant curvature

In this paper, three-dimensional Dunkl oscillator models are studied in a generalized form of superintegrable Euclidean Hamiltonian systems to curved ones. These models are defined based on curved Hamiltonians, which depend on a deformation parameter of underlying space and involve reflection operators. The corresponding symmetries are obtained by the Jordan–Schwinger representations in the family of the Cayley–Klein orthogonal algebras using the creation and annihilation operators of the dynamical sl−1(2) algebra of the one-dimensional Dunkl oscillator. The resulting algebra is a deformation of soκ1κ2(4) with reflections, which is known as the Jordan–Schwinger–Dunkl algebra jsdκ1κ2(4). Hence, it is shown that this model is maximally superintegrable. On the other hand, the superintegrability of the three-dimensional Dunkl oscillator model is studied from the viewpoint of the factorization approach. The spectrum of this system is derived through the separation of variables in geodesic polar coordinates, and the resulting eigenfunctions are algebraically given in terms of Jacobi polynomials.

A representation of the Dunkl oscillator model on curved spaces: Factorization approach

In this paper, we study the Dunkl oscillator model in a generalization of superintegrable Euclidean Hamiltonian systems to the two-dimensional curved ones with a m:n frequency ratio. This defined model of the two-dimensional curved systems depends on a curvature/deformation parameter of the underlying space involving reflection operators. The curved Hamiltonian Hκ admits the separation of variables in both geodesic parallel and polar coordinates, which generalizes the Cartesian coordinates of the plane. Similar to the behavior of the Euclidean case, which is the κ → 0 limit case of the curved space, the superintegrability of a curved Dunkl oscillator is naturally understood from the factorization approach viewpoint in that setting. Therefore, their associated sets of polynomial constants of motion (symmetries) as well as algebraic relations are obtained for each of them separately. The energy spectrum of the Hamiltonian Hκ and the separated eigenfunctions are algebraically given in terms of hypergeometric functions and in the special limit case of null curvature occur in the Laguerre and Jacobi polynomials. Finally, the overlap coefficients between the two bases of the geodesic parallel and polar coordinates are given by hypergeometric polynomials.

The anisotropic Dunkl oscillator problem on the two-dimensional curved spaces

In this paper, we study the two-dimensional (2D) Euclidean anisotropic Dunkl oscillator model in an integrable generalization to curved ones of the 2D sphere [Formula: see text] and the hyperbolic plane [Formula: see text]. This generalized model depends on the deformation parameter [Formula: see text] of underlying space and involves reflection operators [Formula: see text] in such a way that all the results are simultaneously valid for [Formula: see text], [Formula: see text] and [Formula: see text]. It turns out that this system is superintegrable based on the special cases of parameter [Formula: see text], which constant measures the asymmetry of the two frequencies in the 2D Dunkl model. Therefore, the Hamiltonian [Formula: see text] can be interpreted as an anisotropic generalization of the curved Higgs–Dunkl oscillator in the limit [Formula: see text]. When [Formula: see text], the system turns out to be the well-known superintegrable 1:2 Dunkl oscillator on [Formula: see text] and [Formula: see text]. In this way, the integrals of the motion arising from the anisotropic Dunkl oscillator are quadratic in the Dunkl derivatives for the special cases of [Formula: see text]. Moreover, these symmetries obtain by the Jordan–Schwinger representation in the family of the Cayley–Klein orthogonal algebras using the creation and annihilation operators of the dynamical [Formula: see text] algebra of the 1D Dunkl oscillator. The resulting algebra is a deformation of [Formula: see text] with reflections, which is named the Jordan–Schwinger–Dunkl algebra [Formula: see text]. The spectrum of this system is determined by the separation of variables in geodesic polar coordinates, and the resulting eigenfunctions are algebraically given in terms of Jacobi polynomials.

Learning local shape descriptors for computing non-rigid dense correspondence

A discriminative local shape descriptor plays an important role in various applications. In this paper, we present a novel deep learning framework that derives discriminative local descriptors for deformable 3D shapes. We use local “geometry images” to encode the multi-scale local features of a point, via an intrinsic parameterization method based on geodesic polar coordinates. This new parameterization provides robust geometry images even for badly-shaped triangular meshes. Then a triplet network with shared architecture and parameters is used to perform deep metric learning; its aim is to distinguish between similar and dissimilar pairs of points. Additionally, a newly designed triplet loss function is minimized for improved, accurate training of the triplet network. To solve the dense correspondence problem, an efficient sampling approach is utilized to achieve a good compromise between training performance and descriptor quality. During testing, given a geometry image of a point of interest, our network outputs a discriminative local descriptor for it. Extensive testing of non-rigid dense shape matching on a variety of benchmarks demonstrates the superiority of the proposed descriptors over the state-of-the-art alternatives.

New superintegrable models on spaces of constant curvature

A general class of superintegrable systems on 2D spaces of constant curvature is known for which the potential is not spherically symmetric but allows separation of variables in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalized Kepler potential. Unlike the radial parts, the angular ones are given implicitly. In the present paper new two-parameter families of angular potentials are constructed in terms of elementary functions. It is shown that for appropriate choice of parameters a family corresponding to the oscillator or Kepler type radial potential reduces to the Poschl–Teller potential. This allows considering Hamiltonian systems defined by this family as generalizations of Tremblay–Turbiner–Winternitz (TTW) or Post–Winternitz (PW) models, both on the plane and on curved spaces of constant curvature.

Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature.

We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators (-Δ±β 2) in d-dimensional, R-radius hyperbolic and hyperspherical geometry, which represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. In particular, we compute closed-form expressions for fundamental solutions of (-Δ±β 2) on , (-Δ±β 2) on , and present two candidate fundamental solutions for (-Δ - β 2) on . Flat-space limits, with their corresponding asymptotic representations, are used to restrict proportionality constants for these fundamental solutions. In order to accomplish this, we summarize and derive new large degree asymptotics for associated Legendre and Ferrers functions of the first and second kind. Furthermore, we prove that our fundamental solutions on the hyperboloid are unique due to their decay at infinity. To derive Gegenbauer polynomial expansions of our fundamental solutions for Helmholtz operators on hyperspheres and hyperboloids, we derive a collection of infinite series addition theorems for Ferrers and associated Legendre functions which are generalizations and extensions of the addition theorem for Gegenbauer polynomials. Using these addition theorems, in geodesic polar coordinates for dimensions greater than or equal to three, we compute Gegenbauer polynomial expansions for these fundamental solutions, and azimuthal Fourier expansions in two-dimensions.

Parametric Blending of Triangular Meshes

We present a new method for constructing a parametric blending surface that smoothly connects two triangular meshes. A user selects a subregion on each triangular mesh. The local parameterizations of two selected regions are found by using geodesic polar coordinates, and a base surface on each triangular mesh is constructed by two boundary curves on parametric domain. Finally, two base surfaces are smoothly blended for generating a blending surface. The shape of a blending surface can easily be controlled by several shape parameters or by directly manipulating surface point. We demonstrate the effectiveness of our technique by showing several modeling examples.

User controllable anisotropic shape distribution on 3D meshes

This paper presents an automatic method for computing an anisotropic 2D shape distribution on an arbitrary 2-manifold mesh. Our method allows the user to specify the direction as well as the density of the distribution. Using a pre-computed lookup table, our method can efficiently detect collision among the shapes to be distributed on the 3D mesh. In contrast to existing approaches, which usually assume the 2D objects are isotropic and have simple geometry, our method works for complex 2D objects and can guarantee the distribution is conflict-free, which is a critical constraint in many applications. It is able to compute multi-class shape distributions in parallel. Our method does not require global parameterization of the input 3D mesh. Instead, it computes local parameterizations on the fly using geodesic polar coordinates. Thanks to a recent breakthrough in geodesic computation, the local parameterization can be computed at low cost. As a result, our method can be applied to models with complicated geometry and topology. Experimental results on a wide range of 3D models and 2D anisotropic shapes demonstrate the good performance and effectiveness of our method.

Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials.

Design and Reuse of Freeform Features on Mesh Surface

To solve the difficulty of reusing complex freeform features on mesh surface,a detail-preserving method is proposed. By means of differential geometry information contained in Laplacian coordinates,such as curvature,normal and so on,put forward a new method to parameterize the boundary loop of features,which is based on differential coordinates. The method of parameterization based on differential coordinates can preserve the length,curvature and the shape of features' boundary curve,which can reduce the deformation of the boundary curve mapped onto the target mesh. The target mesh is parameterized by Geodesic Polar Coordinates method. The boundary curve of features is mapped onto target mesh by the principle of common parameter coordinates. Thus the boundary constraint conditions including position constraint conditions and normal constrain conditions,which are used as the constrain conditions of mesh deformation; can get from the mapped boundary loop. Finally,the feature is deformed to match the target mesh using Laplacian deformation which combines with linear rotation-invariant coordinates. The algorithm avoids parameterizing the reuse features directly,which makes it easy to work with the complex features,and unrelated with the number of feature genus. The results show that the proposed method is robust,effective and can be used to clone the complex features.

Differential geometric methods for examining the dynamics of slow-fast vector fields

In this work we present computational methods for examining dynamical systems. We focus on those systems being characterized by slow---fast vector fields or corresponding differential algebraic equations that commonly occur in physical applications. In the latter ones scientists usually consider a manifold of admissible physical states and a vector field describing the time evolution of the physical system. The manifold is typically implicitly defined within a higher-dimensional space by a system of equations. Certain physical systems, such as relaxation oscillators, perform sudden jumps in their state evolution when they are forced into an unstable state. The main contribution of the present work is to model the dynamical evolution incorporating the jumping behavior from a perspective of computational geometry which not only provides a qualitative analysis but also produces quantitative results. We use geodesic polar coordinates (GPC) to numerically obtain explicit parametrizations of the implicitly defined manifold and of the relevant jump and hit sets. Moreover, to deal with the possibly high co-dimension of the considered implicitly defined manifold we sketch how GPC in combination with the cut locus concept can be used to numerically obtain an essentially injective global parametrization. This allows us to parametrize and visualize the dynamical evolution of the system including the aforementioned jump phenomena. As main tools we use homotopy approaches in conjunction with concepts from differential geometry. We discuss how to numerically realize and how to apply them to several examples from mechanics, electrical engineering and biology.

Superintegrable systems on spaces of constant curvature

Construction and classification of two-dimensional (2D) superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of constant curvature and separable in the so-called geodesic polar coordinates are presented. The method proposed is applicable to any value of curvature including the case of Euclidean plane, sphere and hyperbolic plane. The main result is a generalization of Bertrand’s theorem on 2D spaces of constant curvature and covers most of the known separable and superintegrable models on such spaces (in particular, the so-called Tremblay–Turbiner–Winternitz (TTW) and Post–Winternitz (PW) models which have recently attracted some interest).

Pattern Formation for the Swift-Hohenberg Equation on the Hyperbolic Plane

In this paper we present an overview of pattern formation analysis for an analogue of the Swift-Hohenberg equation posed on the real hyperbolic space of dimension two, which we identify with the Poincaré disc \(\mathbb{D }\). Different types of patterns are considered: spatially periodic stationary solutions, radial solutions and traveling waves, however there are significant differences in the results with the Euclidean case. We apply equivariant bifurcation theory to the study of spatially periodic solutions on a given lattice of \(\mathbb{D }\) also called H-planforms in reference with the “planforms” introduced for pattern formation in Euclidean space. We consider in details the case of the regular octagonal lattice and give a complete descriptions of all H-planforms bifurcating in this case. For radial solutions (in geodesic polar coordinates), we present a result of existence for stationary localized radial solutions, which we have adapted from techniques on the Euclidean plane. Finally, we show that unlike the Euclidean case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf bifurcation to traveling waves which are invariant along horocycles of \(\mathbb{D }\) and periodic in the “transverse” direction. We highlight our theoretical results with a selection of numerical simulations.

Geodesic spheres and non radial eigenfunctions on Damek–Ricci spaces

Let S=NA be a Damek–Ricci space with its standard metric γS. Let C:S→B be the Cayley transform from S onto the unit ball B in s. We compute the transported metric γB=C−1∗(γS). By separating variables in geodesic polar coordinates, we then compute the non-radial M-invariant eigenfunctions of the Laplacian on S, where M is the group of automorphisms of S preserving the inner product on s. The “radial” part of these eigenfunctions is given by (associated) Jacobi functions. The “angular” part is given by certain orthogonal polynomials in two variables studied by Koornwinder.

Geodesic Polar Coordinates on Polygonal Meshes

AbstractGeodesic Polar Coordinates (GPCs) on a smooth surface S are local surface coordinates that relates a surface point to a planar parameter point by the length and direction of a corresponding geodesic curve onS. They are intrinsic to the surface and represent a natural local parameterization with useful properties. We present a simple and efficient algorithm to approximate GPCs on both triangle and general polygonal meshes. Our approach, named DGPC, is based on extending an existing algorithm for computing geodesic distance. We compare our approach with previous methods, both with respect to efficiency, accuracy and visual qualities when used for local mesh texturing. As a further application we show how the resulting coordinates can be used for vector space methods like local remeshing at interactive frame‐rates even for large meshes.

Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

Due to the isotropy of d-dimensional hyperbolic space, there exists a spherically symmetric fundamental solution for its corresponding Laplace–Beltrami operator. The R-radius hyperboloid model of hyperbolic geometry with R > 0 represents a Riemannian manifold with negative-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace’s equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the hyperbolic sine, finite summation expressions over hyperbolic functions, Gauss hypergeometric functions and in terms of the associated Legendre function of the second kind with order and degree given by d/2 − 1 with real argument greater than unity. We also demonstrate uniqueness for a fundamental solution of Laplace’s equation on this manifold in terms of a vanishing decay at infinity. In rotationally invariant coordinate systems, we compute the azimuthal Fourier coefficients for a fundamental solution of Laplace’s equation on the R-radius hyperboloid. For d ⩾ 2, we compute the Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace’s equation on this negative-constant curvature Riemannian manifold. In three dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace’s equation is obtained through comparison with its corresponding Gegenbauer expansion.

Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature

A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of the motion (besides the Hamiltonian) which are ex- plicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions cover the six spaces in a unified way as these are parametrized by two contraction parameters that govern the curvature and the signature of the metric on each space. Next two maxi- mally superintegrable Hamiltonians are identified within the initial superintegrable family by finding the remaining constant of the motion. The former potential is the superposition of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers (depending on each specific space), so that this generalizes the Smorodinsky-Winternitz sys- tem. The latter one is a superposition of the Kepler-Coulomb potential with another two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz vector for these spaces is deduced. Furthermore both potentials are analysed in detail for each particular space. Some comments on their generalization to arbitrary dimension are also presented.