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Connection of Bézier Curve and its Extension

Author: YangLinYing
Tutor: ZhangGuiCang
School: Northwest Normal University
Course: Applied Mathematics
Keywords: Bézier curves B-spline curves λαβ-Bézier curves NURBS curves Smooth connection
CLC: TP391.7
Type: Master's thesis
Year: 2013
Downloads: 9
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Abstract


The Bézier method is a milepost in the field of curves and surface modeling,it is based on the approximability principle. Applying with Bézier method, we canconveniently approximate the mathematical curves and surfaces, or the sketch is de-lineated by the designer, playing the role of”aided design” genuinely, However, dueto the complex shape of the line and surface in the practical application, it is difcultto use a single curve!surface to describe its appearance perfectly. Therefore, howto smoothy connect curves!surfaces becomes the key to express a complex combi-nation of curves and surfaces. In this thesis, Firstly, A λαβ-Bézier curves is given,the curves can generate diferent curves approximating the unified control polygonby choosing diferent parameter curves, achieving more accurate design for the re-quired curves. Splicing theorem of the curves and the Bézier curves is obtained.Examples of the continuous conditions between λαβ-Bézier curves of diferent pa-rameters and Bézier curves are given. Secondly, utilizing Bézier constructive methodof B-spline curves, using Bézier curves splicing technology to achieve the connectionB-spline curves and Bézier curves. The G0!G1and G2continuity conditions be-tween the cubic Bézier curves and quadratic B-spline curves are obtained. Thirdly,using conversion method to NURBS curves and rational Bézier curves, we achievethe mutual conversion of non uniform B-spline curves and Bézier curves, the prob-lem of connection between cubic Bézier curves and quadratic Non-uniform B-splinecurves is changed into another problem, connection between cubic Bézier curvesand quadratic Bézier curves, it is obtained that the continuous conditions betweenthe cubic Bézier curves and quadratic Non-uniform B-spline curves. Finally, a setof cubic polynomial with one shape parameter is defined, the shape adjustabilityof the curves is discussed, the properties of this set of basis functions is analyzed, the continuous conditions of curves and application examples in curves and surfacesmodeling are presented.

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