Related papers: Barycentric Coordinates as Interpolants
We propose a novel theoretical framework for barycentric interpolation, using concepts recently developed in mathematical physics. Generalized barycentric coordinates are defined similarly to Shepard's method, using positive geometries -…
We propose a variational technique to optimize for generalized barycentric coordinates that offers additional control compared to existing models. Prior work represents barycentric coordinates using meshes or closed-form formulae, in…
Spherical and polar geometries arise in many important areas of computational science, including weather and climate forecasting, optics, and astrophysics. In these applications, tensor-product grids are often used to represent unknowns.…
Each point of a simplex is expressed as a unique convex combination of the vertices. The coefficients in the combination are the barycentric coordinates of the point. For each point in a general convex polytope, there may be multiple…
We present a novel barycentric interpolation algorithm designed for analytic functions $f\in\mathcal{A}(E)$ defined on the complex plane. The algorithm, which encompasses both polynomial and rational interpolation, is tailored to handle…
Barycentric coordinates provide solutions to the problem of expressing an element of a compact convex set as a convex combination of a finite number of extreme points of the set. They have been studied widely within the geometric…
Optical surfaces represented by second-degree polynomials (quadratic or conics) are ubiquitous in optics. We revisit the equations of the conic shapes in the context of grazing incidence optics, gathering together the curves commonly used…
Image interpolation, or image morphing, refers to a visual transition between two (or more) input images. For such a transition to look visually appealing, its desirable properties are (i) to be smooth; (ii) to apply the minimal required…
The Cartesian coordinate system is the most commonly used system in computer visualization. This is due to its ease of use and processing speed. However, it is not always suitable for a given problem. Angular measures often allow us to…
In this paper, applied strictly monotonic increasing scaled maps, a kind of well-conditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as $x^\alpha$ for $\alpha \in…
This paper presents a convenient graphical tool for encoding visual patterns (such as image patches and image atoms) as point constellations in a space spanned by perceptual features and with a clear geometrical interpretation. General…
This paper presents a unified metric-based framework for triangle geometric inequalities using barycentric coordinates. By interpreting classical inequalities as squared distances between points(a process termed metricization)we derive and…
Multi-sided surfaces are often defined by side interpolants (also called ribbons), i.e. the surface has to connect to the ribbons with a prescribed degree of smoothness. The I-patch is such a family of implicit surfaces capable of…
Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the…
Birefringent crystals are extensively used to manipulate polarized light. The generalized transfer matrix developed allows efficient calculation of the full polarization state of light transmitted through and reflected by a stack of…
This paper describes a novel construction of generalized barycentric coordinates of points on a sphere with respect to the vertices of a given spherical polygon that is contained in a common hemisphere. While in the standard approach such…
We examine implicit representations of parametric or point cloud models, based on interpolation matrices, which are not sensitive to base points. We show how interpolation matrices can be used for ray shooting of a parametric ray with a…
Astronomical data does not always use Cartesian coordinates. Both all-sky observational data and simulations of rotationally symmetric systems, such as accretion and protoplanetary discs, may use spherical polar or other coordinate systems.…
Diffractive lenses fabricated by lithographic methods are one of the most popular image forming optics in the x-ray regime. Most commonly, binary diffractive optics, such as Fresnel zone plates are used due to their ability to focus at high…
We describe some new univariate spline quasi-interpolants on uniform partitions of bounded intervals. Then we give some applications to numerical analysis: integration, differentiation and approximation of zeros.