Abstract
In this paper we study the curvature term for the Lagrange and PPH (Amat et al. in Found. Comput. Math. 6:193–225, 2006 and Ortiz and Trillo in Preprint, arXiv:1811.10566, 2018) reconstruction operators in uniform and nonuniform meshes. We also make a comparison between both curvature terms in order to obtain an inequality which clearly shows that the PPH reconstruction presents a lower curvature term. Presenting a low curvature term is crucial in some applications, such as smoothing splines.
Highlights
Reconstruction and subdivision operators have been studied, analyzed and implemented in computer aided geometric design, giving rise to interesting applications in different fields of science
In order to write pL(x) in the form of polynomial (2), we look for the appropriate values of the parameters A and B, which allow for the remaining interpolation conditions to be satisfied: pL(xj–1) = fj–1, pL(xj+2) = fj+2
3 Study of the curvature term in nonuniform meshes Let us consider the set of points fj–1, fj, fj+1, fj+2 corresponding to subsequent ordinates at the abscissas xj–1, xj, xj+1, xj+2 of a nonuniform mesh X
Summary
Reconstruction and subdivision operators have been studied, analyzed and implemented in computer aided geometric design, giving rise to interesting applications in different fields of science. We study the curvature term of the functional (1) for the Lagrange and PPH reconstructions, in the uniform and nonuniform cases. In order to build the pH (x) in the form of polynomial (2), we need to impose the following two conditions: pH (xj–1) = fj–1, pH (xj+2) = fj+2, (10)
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