A planar cubic Bézier spiral
type: Item
instance of: scholarly article
title: A planar cubic Bézier spiral
zbMATH Open document ID: 0857.65019
DOI: 10.1016/0377-0427(95)00246-4
author: D. J. Walton
author: D. S. Meek
published in: Journal of Computational and Applied Mathematics
publication date: 1997-02-20T00:00:00Z
review text: Traditionally, the smooth connection between different curves (e.g., the connection between straight and curved portions of a railway track where continuous curvature transition is of the essence) has been done by transcendental curves. In order to apply CAD methods to construction problems of the aforementioned type, it would be useful to have Bézier curves to replace clothoid curves and circular involutes. The authors give the needed tool in constructing a spiral arc (i.e., an arc of monotone curvature) given by a starting point \({\mathbf B}\), starting and ending tangent vectors \({\mathbf T}\), \({\mathbf T}_1\), making an angle \(\theta\), and a final curvature value \(c\). Define the two parameters \(a=25\tan\theta/54c\cos\theta\), \(b=5\tan\theta/9c\). Then the cubic Bézier curve \[ {\mathbf Q}(u)= {\mathbf B}(1- u)^3+3{\mathbf B}_1(1-u)^2u+ 3{\mathbf B}_2(1-u) u^2+{\mathbf B}_3 u^3,\quad 0\leq u\leq1, \] where \({\mathbf B}_1={\mathbf B}+a{\mathbf T}\), \({\mathbf B}_2={\mathbf B}_1+a{\mathbf T}\), \({\mathbf B}_3={\mathbf B}_2+b{\mathbf T}_1\), is a spiral arc having curvature \(0\) at \({\mathbf B}\) \((u=0)m\), curvature \(c\) at \({\mathbf B}_3\), and tangent vectors \({\mathbf T}\), \({\mathbf T}_1\) at the endpoints. The authors work out all the needed formulas for connections of straight line to circle, circle to circle in all cases of relative positions, and straight line to straight line. This is a very useful paper.
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 65D10
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 933155
zbMATH Keywords: planar cubic Bézier spiral
zbMATH Keywords: continuous curvature transition
zbMATH Keywords: CAD methods
zbMATH Keywords: Bézier curves
zbMATH Keywords: clothoid curves
zbMATH Keywords: circular involutes
zbMATH Keywords: spiral arc
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00246-4
OpenAlex ID: W2038854446
cites work: Q4039899
cites work: Q4732025
cites work: Q5850097
cites work: The use of Cornu spirals in drawing planar curves of controlled curvature
cites work: Shape determination of planar uniform cubic B-spline segments
cites work: A geometric characterization of parametric cubic curves
cites work: Offset curves of clothoidal splines
Recommended article: A further generalisation of the planar cubic Bézier spiral
Recommended article: Approximation of a planar cubic Bézier spiral by circular arcs
Recommended article: Q5701899
Recommended article: Planar \(G^{2}\) transition curves composed of cubic Bézier spiral segments
Recommended article: A New Planar Bezier-Like Spiral Defined by Exponential Functions And Transition Curves
Recommended article: Q5137468
Recommended article: Spiral arc spline approximation to a planar spiral
Recommended article: Q5446807
Recommended article: Offset-rational sinusoidal spirals in bézier form
description: scientific article; zbMATH DE number 933155