Convexity preserving splines over triangulations
type: Item
instance of: scholarly article
title: Convexity preserving splines over triangulations
zbMATH Open document ID: 1229.65043
author: Larry L. Schumaker
author: Hendrik Speleers
published in: Computer Aided Geometric Design
publication date: 2011-07-20T00:00:00Z
full work available at URL: https://lirias.kuleuven.be/handle/123456789/267195
review text: Visualization of 3D data in the form of a surface view is one of the important topics in Computer Graphics. The problem gets critically important when the data possesses some inherent shape feature. This paper is concerned with the solution of this problem when the data has a convex shape and its visualization is required to have similar inherent features to that of the data, and its purpose is twofold. First, to describe a general method for constructing sets of sufficient linear conditions in terms of the control points of a triangular Bézier surface that ensure convexity. The method, which is based on blossoming, can be used to create nested sequences of weaker and weaker conditions, including all of the known sets of linear conditions. Second, to develop explicit convexity preserving interpolation and approximation methods using the linear convexity conditions together with certain macro-element spline spaces defined on triangulations.
reviewed by: Juan Monterde
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 65D18
zbMATH DE Number: 5929152
zbMATH Keywords: spline interpolation
zbMATH Keywords: shape preservation
zbMATH Keywords: convex surfaces
zbMATH Keywords: visualization of 3D data
zbMATH Keywords: computer graphics
zbMATH Keywords: triangular Bézier surface
zbMATH Keywords: blossoming
zbMATH Keywords: triangulations
MaRDI profile type: Publication
OpenAlex ID: W2058468256
cites work: A geometric criterion for the convexity of Powell-Sabin interpolants and its multivariate generalization
cites work: The convexity of quadratic parametric triangular Bernstein-Bézier surfaces
cites work: Multivariate convexity preserving interpolation by smooth functions
cites work: Convexity preserving interpolation and Powell-Sabin elements
cites work: Piecewise linear interpolants to Lagrange and Hermite convex scattered data
cites work: Linear convexity conditions for rectangular and triangular Bernstein-Bézier surfaces
cites work: Convexity preserving scattered data interpolation using Powell-Sabin elements
cites work: The convexity of Bernstein polynomials over triangles
cites work: An improved condition for the convexity of Bernstein-Bézier surfaces over triangles
cites work: Convexity of parametric Bézier surfaces in terms of Gaussian curvature signatures
cites work: Q3975818
cites work: Q3031411
cites work: On calculating normalized Powell-Sabin B-splines
cites work: Algorithms for surface fitting using Powell-Sabin splines
cites work: The invariance of weak convexity conditions of \(B\)-nets with respect to subdivision
cites work: A weak condition for the convexity of tensor-product Bézier and B- spline surfaces
cites work: A counterexample to a theorem about the convexity of Powell--Sabin elements
cites work: Convexity of Bézier nets on triangulations
cites work: Bézier nets, convexity and subdivision on higher-dimensional simplices
cites work: On convexity of piecewise polynomial functions on triangulations
cites work: Convexity of Bézier nets on sub-triangles
cites work: Shape criteria of Bernstein-Bézier polynomials over simplexes
cites work: Surface fitting using convex tensor-product splines
cites work: Q3839110
cites work: Arbitrarily weak linear convexity conditions for multivariate polynomials
cites work: Convexity conditions for parametric tensor-product B-spline surfaces
cites work: Q4291349
cites work: Convex preserving scattered data interpolation using bivariate \(C^1\) cubic splines
cites work: Spline Functions on Triangulations
cites work: C1 Convexity-Preserving Interpolation of Scattered Data
cites work: Convexity preserving interpolation
cites work: The conditions of convexity for Bernstein-Bézier surfaces over triangles
cites work: Q4257514
cites work: On the convexity of \(C^1\) surfaces associated with some quadrilateral finite elements
cites work: On the convexity of Bézier nets of quadratic Powell-Sabin splines on 12-fold refined triangulations
cites work: Multivariate Bernstein polynomials and convexity
cites work: Computing bivariate splines in scattered data fitting and the finite-element method
cites work: An improved condition for the convexity and positivity of Bernstein- Bézier surfaces over triangles
cites work: Surface fitting using convex Powell-Sabin splines
cites work: The convexity of parametric Bézier triangular patches of degree 2
cites work: On the convexity of parametric Bézier triangular surfaces
DOI: 10.1016/J.CAGD.2011.03.001
description: scientific article; zbMATH DE number 5929152