Geometric Algorithms And Their Applications Of C-Bézier Curves In CAD

In CAD,modeling an integral curve often brings redundant data.In fact,when modeling a whole curve,the control points of different segments may have relations between each other.To find the relations,this thesis chooses to study C-Bézier curves.In addition,the shape of the C-Bézier curve is adjusted by its control points and the C-Bézier curve is the trajectory of a point turning around the center of an ellipse orbit while the orbit plane is moving with the center along a Bézier curve.Based on the geometric shape of C-Bézier curve,a new method is proposed to adjust its shape by geometric characters.Thus,this thesis from the theory and application viewpoint,profoundly studies the geometric algorithms for the C-Bézier curves in CAD and pays attention to discuss two parts-migration algorithm and shape adjusting algorithm.The research achievements and main contents are as follows:1.The migration algorithm of C-Bézier curve is studied.There often exists redundant datas when constructing geometric model in CAD.To solve this problem,the non-negative interval of the traditional C-Bézier curve is extended,and its interval is defined as the whole real number field.Then a new integral C-Bézier is constructed,which is made up of the traditional inner segment in a finite closed interval(such as[0,a])and the part out of the interval.The focus of this thesis is to consider the change of control vertices for the C-Bézier curve when the original parametric region[0,a]is scaled on(-?,+?).By comparing with basis functions recursively and looking for relations of control vertices before and after migration,we can draw a conclusion that motion curves can be represented as an inner C-Bézier form as long as its parameter interval length is in(0,?).What's more,the new produced control points can be obtained by a direct linear combination or stepwise linear interpolation form of the old ones.Considering integral curve and migration algorithm may be used to scale intervals of modeling and reduce redundant.2.The shape adjustment algorithm of C-Bézier curve is studied.In CAD,the shape of a free formed curve is often adjusted by its control points.In this thesis,we adjust the curve shape by some geometric characters.We focus on any C-Bézier curve,which has been known as a path of a point doing two types of movements,moving along a Bézier curve and rotating in an elliptical orbit,at the same time.Given the geometric characters(two half-axis vectors of an ellipse,the corresponding angles of the start and end points of the curve,and the control points of the center Bézier curve),the explicit formulas of C-Bézier curve are obtained by calculating the angle of rotation in separation form and relation matrix among control points from Bézier and triangular part to the C-Bézier curves.And at last,the impacts of the geometric characters on the shapes of the C-Bézier curve are analyzed from three aspects:uniform scaling the elliptical radius vectors,given angles and moving control vertices of its center Bézier curve.