Planar interpolation with a pair of rational spirals
type: Item
instance of: scholarly article
title: Planar interpolation with a pair of rational spirals
zbMATH Open document ID: 1112.41001
author: D. S. Meek
author: Tim N. T. Goodman
published in: Journal of Computational and Applied Mathematics
publication date: 2007-03-08T00:00:00Z
review text: \(G^2\) Hermite data is defined as follows: (i) a pair of points \(A_0\) and \(A_1\), (ii) unit tangent vectors \(T_{0},T_{1}\) at \(A_0\) and \(A_1\), and a (iii) pair of signed curvatures \(\kappa_0,\kappa_1\) at \(A_0,A_1\). This paper addresses the \(G^2\) Hermite interpolation problem of finding a curve that fits the Hermite data when the data is based on a finite set of points at which the curvature vanishes or when the curvature changes sign between consecutive points. In the case of zero curvature at the ends, one seeks a C-shaped solution by adjoining two spiral segments; in the case of curvature changing sign one seeks an S-shaped solution by adjoining a pair of spiral segments. This paper addresses the feasibility of finding a suitable pair of spirals depending on the Hermite data. A spiral is defined as a curve of monotone increasing or monotone decreasing curvature that does not change sign. In the case of vanishing curvature at \(A_0\) and \(A_1\), let \(B\) denote the point at which the spiral segments are joined and let \(\alpha_i\), \(i=0,1\) denote the external angle between the line segment joining \(A_0,A_1\) and the tangent line at \(A_i\). Finding a C-shaped pair of spiral segments that fits the data depends on the location of \(B\) and the sum of the angles \(\alpha_i\). Several diagrams are provided to describe precisely the conditions required on the location of \(B\) depending on whether \(\alpha_1+\alpha_2\) is less than \(\pi\), equal to \(\pi \) or greater than \(\pi\). In the case \(\alpha_1+\alpha_2<\pi\) the condition says, roughly, that \(B\) should be near the centroid of the triangle formed by the line segment joining \(A_0,A_1\) and the tangent lines at \(A_i\). In the case of curvature changing sign, the feasible region for \(B\) is described in terms of the circles \(C_i\) of curvature at the \(A_i\) and the tangent lines to the \(C_i\) that are perpendicular to the tangents \(T_i\). The equation generating the spirals from whic…
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 5131547
zbMATH Keywords: planar interpolation
zbMATH Keywords: rational spirals
zbMATH Keywords: Hermite interpolation
reviewed by: Joseph Lakey
MaRDI profile type: Publication
full work available at URL: j.cam.2006.02.003
OpenAlex ID: W2156198421
cites work: Q4256595
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cites work: Q5725219
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cites work: Clothoid Spline Transition Spirals
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DOI: 10.1016/J.CAM.2006.02.003
description: scientific article; zbMATH DE number 5131547