Smooth polynomial approximation of spiral arcs

Authors:
R. J. Cripps;M. Z. Hussain;S. Zhu
Affiliations:
Geometric Modelling Group, Mechanical and Manufacturing Engineering, University of Birmingham, UK;Geometric Modelling Group, Mechanical and Manufacturing Engineering, University of Birmingham, UK;Geometric Modelling Group, Mechanical and Manufacturing Engineering, University of Birmingham, UK
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

Citing 7
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Abstract

Constructing fair curve segments using parametric polynomials is difficult due to the oscillatory nature of polynomials. Even NURBS curves can exhibit unsatisfactory curvature profiles. Curve segments with monotonic curvature profiles, for example spiral arcs, exist but are intrinsically non-polynomial in nature and thus difficult to integrate into existing CAD systems. A method of constructing an approximation to a generalised Cornu spiral (GCS) arc using non-rational quintic Bezier curves matching end points, end slopes and end curvatures is presented. By defining an objective function based on the relative error between the curvature profiles of the GCS and its Bezier approximation, a curve segment is constructed that has a monotonic curvature profile within a specified tolerance.