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Università della Svizzera italiana

New shape control tools for rational Bézier curve design

Ramanantoanina, Andriamahenina ; Hormann, Kai

In: Computer aided geometric design, 2021, vol. 88, p. 11

Bézier curves are indispensable for geometric modelling and computer graphics. They have numerous favourable properties and provide the user with intuitive tools for editing the shape of a parametric polynomial curve. Even more control and flexibility can be achieved by associating a shape parameter with each control point and considering rational Bézier curves, which comes with the...

Università della Svizzera italiana

On the Lebesgue constant of barycentric rational Hermite interpolants at equidistant nodes

Cirillo, Emiliano ; Hormann, Kai

In: Journal of computational and applied mathematics, 2019, vol. 349, p. 292-301

Barycentric rational Floater–Hormann interpolants compare favourably to classical polynomial interpolants in the case of equidistant nodes, because the Lebesgue constant associated with these interpolants grows logarithmically in this setting, in contrast to the exponential growth experienced by polynomials. In the Hermite setting, in which also the first derivatives of the interpolant are...

Università della Svizzera italiana

Behaviour of exponential three-point coordinates at the vertices of convex polygons

Anisimov, Dmitry ; Hormann , Kai ; Schneider

In: Journal of computational and applied mathematics, 2019, vol. 350, p. 114-129

Barycentric coordinates provide a convenient way to represent a point inside a triangle as a convex combination of the triangle’s vertices and to linearly interpolate data given at these vertices. Due to their favourable properties, they are commonly applied in geometric modelling, finite element methods, computer graphics, and many other fields. In some of these applications, it is desirable...

Università della Svizzera italiana

Advances in barycentric rational interpolation of a function and its derivatives

Cirillo, Emiliano ; Hormann, Kai (Dir.)

Thèse de doctorat : Università della Svizzera italiana, 2019 ; 2019INFO007.

Linear barycentric rational interpolants are a particular kind of rational interpolants, defined by weights that are independent of the function f. Such interpolants have recently proved to be a viable alternative to more classical interpolation methods, such as global polynomial interpolants and splines, especially in the equispaced setting. Other kinds of interpolants might indeed suffer...

Università della Svizzera italiana

Subdivision schemes for curve design and image analysis

Volontè, Elena ; Hormann, Kai (Dir.) ; Rossini, Milvia (Codir.)

Thèse de doctorat : Università della Svizzera italiana, 2018 ; 2018INFO001.

Subdivision schemes are able to produce functions, which are smooth up to pixel accuracy, in a few steps through an iterative process. They take as input a coarse control polygon and iteratively generate new points using some algebraic or geometric rules. Therefore, they are a powerful tool for creating and displaying functions, in particular in computer graphics, computer-aided design, and...

Università della Svizzera italiana

Theory and applications of bijective barycentric mappings

Schneider, Teseo ; Hormann, Kai (Dir.)

Thèse de doctorat : Università della Svizzera italiana, 2017 ; 2017INFO007.

Barycentric coordinates provide a convenient way to represent a point inside a triangle as a convex combination of the triangle's vertices, and to linearly interpolate data given at these vertices. Due to their favourable properties, they are commonly applied in geometric modelling, finite element methods, computer graphics, and many other fields. In some of these applications it is desirable...

Università della Svizzera italiana

Analysis and new constructions of generalized barycentric coordinates in 2D

Anisimov, Dmitry ; Hormann, Kai (Dir.)

Thèse de doctorat : Università della Svizzera italiana, 2017 ; 2017INFO004.

Different coordinate systems allow to uniquely determine the position of a geometric element in space. In this dissertation, we consider a coordinate system that lets us determine the position of a two-dimensional point in the plane with respect to an arbitrary simple polygon. Coordinates of this system are called generalized barycentric coordinates in 2D and are widely used in computer...

Università della Svizzera italiana

Efficient multi-bounce lightmap creation using GPU forward mapping

Schärfig, Randolf ; Hormann, Kai (Dir.) ; Stamminger, Marc (Codir.)

Thèse de doctorat : Università della Svizzera italiana, 2016 ; 2016INFO008.

Computer graphics can nowadays produce images in realtime that are hard to distinguish from photos of a real scene. One of the most important aspects to achieve this is the interaction of light with materials in the virtual scene. The lighting computation can be separated in two different parts. The first part is concerned with the direct illumination that is applied to all surfaces lit by a...

Université de Fribourg

Barycentric rational interpolation at quasi-equidistant nodes

Hormann, Kai ; Klein, Georges ; De Marchi, Stefano

In: Dolomites Research Notes on Approximation, 2012, vol. 5, no. 1, p. 1-6

A collection of recent papers reveals that linear barycentric rational interpolation with the weights suggested by Floater and Hormann is a good choice for approximating smooth functions, especially when the interpolation nodes are equidistant. In the latter setting, the Lebesgue constant of this rational interpolation process is known to grow only logarithmically with the number of nodes....

Université de Fribourg

On the Lebesgue constant of barycentric rational interpolation at equidistant nodes

Bos, Len ; De Marchi, Stefano ; Hormann, Kai ; Klein, Georges

In: Numerische Mathematik, 2012, vol. 121, p. 461–471

Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than their polynomial analogue. A natural and important...

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17 Hormann, Kai