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About 8,144 papers for “triangular surfaces”

Triangular bubble spline surfaces

Mario Kapl, Marek Byrtus, Bert Jüttler · Computer-Aided Design · 2011

We present a new method for generating a [Formula: see text]-surface from a triangular network of compatible surface strips. The compatible surface strips are given by a network of polynomial curves with an associated implicitly defined surface, which fulfill certain compatibility conditions. Our construction is based on a new concept, called bubble patches, to represent the single surface patches. The compatible surface strips provide a simple [Formula: see text]-condition between two neighboring bubble patches, which are used to construct surface patches, connected with [Formula: see text]-continuity. For [Formula: see text], we describe the obtained [Formula: see text]-condition in detail. It can be generalized to any [Formula: see text]. The construction of a single surface patch is based on Gordon-Coons interpolation for triangles.Our method is a simple local construction scheme, which works uniformly for vertices of arbitrary valency. The resulting surface is a piecewise rational surface, which interpolates the given network of polynomial curves. Several examples of [Formula: see text], [Formula: see text] and [Formula: see text]-surfaces are presented, which have been generated by using our method. The obtained surfaces are visualized with reflection lines to demonstrate the order of smoothness.

Cited by 5Open access

Multiresolution triangular B-spline surfaces

A. Dreger, M.H. Gross, Jürgen Schlegel · 2002

We present multiresolution B-spline surfaces of arbitrary order defined over triangular domains. Unlike existing methods, the basic idea of our approach is to construct the triangular basis functions from their tensor product relatives in the spirit of box splines by projecting them into the barycentric plane. The scheme works for splines of any order where the fundamental building blocks of the surface are hierarchies of triangular B-spline scaling functions and wavelets spanning the complement spaces between levels of different resolution. Although our decomposition and reconstruction schemes operate in principle on a tensor product grid in 3D, the sparsity of the arrangement enables us to design efficient linear time algorithms. The resulting basis functions are used to approximate triangular surfaces and provide many useful properties, such as multiresolution editing, local level of detail, continuity control, surface compression and much more. The performance of our approach is illustrated by various examples including parametric and nonparametric surface editing and compression.

Cited by 6

Subdivision depth for triangular surfaces

Ghulam Mustafa, M.S. Hashmi, Faheem Khan · Alexandria Engineering Journal · 2016

The aim of this attempt was to present an efficient algorithm for the evaluation of error bound of triangular subdivision surfaces. The error estimation technique is based on first order difference and this process is independent of parametrization. This technique can be easily generalized to higher arity triangular surfaces. The estimated error bound is expressed in-terms of initial control point sequence and constants. Here, we efficiently estimate error bound between triangular surface and its control polygon after k-fold subdivision and further extended to evaluate subdivision depth of the scheme.

Cited by 3Open access

Multiresolution triangular B-spline surfaces

A. Dreger, Markus Groß, Jürgen Schlegel · Repository for Publications and Research Data (ETH Zurich) · 1997

We present multiresolution B-spline surfaces of arbitrary order defined over triangular domains. Unlike existing methods, the basic idea of our approach is to construct the triangular basis functions from their tensor product relatives in the spirit of box splines by projecting them into the barycentric plane. The scheme works for splines of any order where the fundamental building blocks of the surface are hierarchies of triangular B-spline scaling functions and wavelets spanning the complement spaces between levels of different resolution. Although our decomposition and reconstruction schemes operate in principle on a tensor product grid in 3D, the sparsity of the arrangement enables us to design efficient linear time algorithms. The resulting basis functions are used to approximate triangular surfaces and provide many useful properties, such as multiresolution editing, local level of detail, continuity control, surface compression and much more. The performance of our approach is illustrated by various examples including parametric and nonparametric surface editing and compression.

Cited by 1Open access

Constrained approximation of rational triangular B\\'ezier surfaces by\n polynomial triangular B\\'ezier surfaces

Stanisław Lewanowicz, Paweł Keller, Paweł Woźny · arXiv (Cornell University) · 2015

We propose a novel approach to the problem of polynomial approximation of\nrational B\\'ezier triangular patches with prescribed boundary control points.\nThe method is very efficient thanks to using recursive properties of the\nbivariate dual Bernstein polynomials and applying a smart algorithm for\nevaluating a collection of two-dimensional integrals. Some illustrative\nexamples are given.\n

Cited by 0Open access

Translation covers among triangular billiard surfaces

Jason Schmurr · Illinois Journal of Mathematics · 2014

We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of triangular billiard surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers.

Cited by 1Open access

Translation Covers Among Triangular Billiards Surfaces

Jason Schmurr · 2008

We identify all translation covers among triangular billiards surfaces. Our main tools are the J-invariant of Kenyon and Smillie [8] and a property of triangular billiards surfaces, which we call fingerprint type, that is invariant under balanced translation covers. 1 1

Cited by 0

Curvature detail representation of triangular surfaces

Jinzhong Wu, Xuehui Liu, Enhua Wu · Journal of Zhejiang University. Science A · 2006

Curvature tells much about details of surfaces and is studied widely by researchers in the computer graphics community. In this paper, we first explain the mean-curvature view of Dirichlet energy of triangular surfaces and introduce a curvature representation of details, and then present surfaces editing applications based on their curvature representation. We apply our method to surfaces with complex boundaries and rich details. Results show the validity and robustness of our method and demonstrate curvature map can be a helpful surfaces detail representation.

Cited by 0

View Factor Calculations Between Triangular Surfaces

Yanan Camaraza · Engineering Perspective · 2025

In radiative heat transfer calculations, to determine the view factor between surfaces is crucial.Currently, the available technical literature on the subject lacks an analytical expression for estimating the view factor for combinations of triangular surfaces.An analytical solution requires the summation of multiple integrals, given the changes in the integration contours, which becomes more complex in irregular contours.This work aims to develop an expression for computing the view factor between 32 triangular geometric configurations with a common edge and included angle .Very importantly, in modern engineering, mesh generators rarely use rectangles or squares (unless the overall geometry is a perfect cube), with triangular elements being the most commonly used elements.For comparison, 48 cases with diverse geometric relationships were calculated using the analytical solution (AS), numerical integration using Simpson's multiple 1/3 rule (MSR), the Sauer graphical solution (SGS), and Bretzhtsov cross roots (BCR).From eight basic geometries, the view factor was computed for the remaining 24 combinations using the sum rule.In all cases, identical fit values were obtained for MSR and SGS with respect to AS, while BCR showed the best correlation in all cases examined.In all the cases evaluated, the BCR showed the best fits, with an error of 6% in more than 90% of the sample s, while the MSR showed an average dispersion of 6% in 65% of the data.Given the practical nature of the contribution and the reasonable values of the fits obtained, the current proposal constitutes a suitable tool for application in thermal engineering.

Cited by 2Open access

New Algorithm of Polynomial Triangular B-B Surfaces Approximation to Rational Triangular B-B Surfaces ★

Huixia Xu A, Guojin Wang A · 2013

In this paper, a new algorithm of approximating rational triangular B-B surfaces using polynomial triangular B-B surfaces is presented. This algorithm has a simple expression and insured convergence. The core idea of the algorithm is to yield a polynomial triangular B-B surface, which has the same degree and new control vertices of the degree-elevated rational triangular B-B surface, to approximate the original rational triangular B-B surface. Then arbitrary given order of the polynomial triangular B-B surface’s hodograph surface necessarily converges uniformly to that of the original rational triangular B-B surface.

Cited by 0