Smoothing of curves and surfaces
type: Item
instance of: scholarly article
title: Smoothing of curves and surfaces
zbMATH Open document ID: 0576.65004
DOI: 10.1016/0167-8396(85)90012-3
author: J. Hoschek
published in: Computer Aided Geometric Design
publication date: 1985-01-01T00:00:00Z
review text: This paper is concerned with the representation of surfaces with a desired shape. When using Bézier or B-splines techniques for interpolation or approximation of curves or surfaces, undesired inflexion points or changes of sign in the Gaussian curvatures may occur. A procedure is described that permits to detect such points by inspecting the associated orthotomic curve or surface. If we consider a point P, not in the surface X, as a light source, the orthotomic surface can be thought as the wave front reflected by X. It turns out that changes in sign in the Gaussian curvatures of X correspond to cusps in the orthotomic surface (caustics), that can be made much more apparent than the changes of curvature in the original surface, as is shown graphically in various examples. It is also indicated how to smooth interactively such undesirable features on Bézier curves by collapsing two of these changes of curvature into a flat point; this is done by finding the zeroes of an appropriate Bézout determinant. This smoothing procedure does not look too practical for surfaces, since fairly high degree polynomials arise in the computation.
reviewed by: Victor Pereyra
Mathematics Subject Classification ID: 65D10
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 3921799
zbMATH Keywords: inflection points
zbMATH Keywords: Bézout-method
zbMATH Keywords: Bézier
zbMATH Keywords: B-splines
zbMATH Keywords: Gaussian curvatures
zbMATH Keywords: orthotomic surface
MaRDI profile type: Publication
full work available at URL: 0167-8396(85)90012-3
OpenAlex ID: W2033558838
cites work: Caustics through the looking glass
cites work: Implicitization, inversion, and intersection of planar rational cubic curves
cites work: Detecting regions with undesirable curvature
cites work: Q5552044
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description: scientific article; zbMATH DE number 3921799