# Controlling the interpolation of NURBS curves and surfaces

Lockyer, Peter Stephen (2007). Controlling the interpolation of NURBS curves and surfaces. University of Birmingham. Ph.D.

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## Abstract

The primary focus of this thesis is to determine the best methods for controlling the interpolation of NURBS curves and surfaces. The various factors that affect the quality of the interpolant are described, and existing methods for controlling them are reviewed. Improved methods are presented for calculating the parameter values, derivative magnitudes, data point spacing and twist vectors, with the aim of producing high quality interpolants with minimal data requirements.

A new technique for obtaining the parameter values and derivative magnitudes is evaluated, which constructs a C$$^1$$ cubic spline with orthogonal first and second derivatives at specified parametric locations. When this data is used to create a C$$^2$$ spline, the resulting interpolant is superior to those constructed using existing parameterisation and derivative magnitude estimation methods.

Consideration is given to the spacing of data points, which has a significant impact on the quality of the interpolant. Existing methods are shown to produce poor results with curves that are not circles. Three new methods are proposed that significantly reduce the positional error between the interpolant and original geometry.

For constrained surface interpolation, twist vectors must be estimated. A method is proposed that builds on the Adini method, and is shown to have improved error characteristics. In numerical tests, the new method consistently outperforms Adini.

Interpolated surfaces are often required to join together smoothly along their boundaries. The constraints for joining surfaces with parametric and geometric continuity are discussed, and the problem of joining $$N$$ patches to form an $$N$$-sided region is considered. It is shown that regions with odd $$N$$ can be joined with G$$^1$$ continuity, but those with even $$N$$ or requiring G$$^2$$ continuity can only be obtained for specific geometries.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Cripps, RobertUNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Schools (1998 to 2008) > School of Engineering
School or Department: Mechanical and Manufacturing Engineering
Funders: Engineering and Physical Sciences Research Council
Subjects: T Technology > TJ Mechanical engineering and machinery
URI: http://etheses.bham.ac.uk/id/eprint/6502

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