Research On Shape Modification And Approximate Degree Reduction Of ?-Bézier Curve

As a research subject of CAD/CAM system,designs of freeform curves and surfaces have always played a vital role,among them Bezier modeling has already become one of powerful tools for describing shape because of its many advantage.However,with the improvement of modern geometric modeling,traditional Bezier have been already difficult to meet demands of practical application.Then,in order to satisfy practical demands,?-Bézier curve has shape adjustable property whilst inheriting the properties of traditional Bezier curve as a new kind of curve modeling.In contrast to no algebraic polynomial modelings,A-Bezier modeling has features such as often resulting in a simple algorithm that is shorter,faster and has easier chosen a parameter,?-Bézier modeling can be used to g enerate useful geometric models.The research contents of this dissertation are organized as follows:(1)The definition and properties of ?-Bézier curve have been summarized in detail.Then,the geometric significance of shape parameter ? has been analyzed emphatically based on properties of ?-Bézier curve,transformational relationship has been derived between ?-Bézier curve of degree n and traditional Bezier curve of degree n+2.In addition,?-Bézier tensor product surface has been defined.(2)Firstly,the definition of allowable interpolation region D of ?-Bézier curve has been obtained,algorithm of shape modification of ?-Bézier curve has been presented by its allowable interpolation region,principle and steps of algorithm has been given.Then,the shape modification of ?-Bézier curve has been investigated for constrained optimizations of a single parameter and multiple parameters separately.Based on Lagrange multiplier method,the shape modification of ?-Bézier curve has been implemented by optimizing m-l+1 perturbations ?i of control points.Using this algorithm,?-Bézier curve has been modified to satisfy the specified constraints of position vector and tangent vector and satisfy the shape-preserving property.Finally,practical examples have been illustrated,results of examples demonstrate that the proposed method of this paper can modify the shape of ?-Bézier curve.(3)For problem of approximate degree reduction of ?-Bézier curve,the method has been presented by the least squared distance.The expression of control points of degree reduction of?-Bézier curve have been confirmed by solving linear equations of satisfying least squared distance,and approximate degree reduction of ?-Bézier curve has been implemented in unrestricted condition,C0 and C1 constraint conditions.At the same time,applied examples and errors of approximate degree reduction have been showed that this algorithm of approximate degree reduction offers a number of new effective means.