Spline Curves Based On Lupas Q-analogue Of Bernstein Operator

In recent years,in computer aided geometric design,the smooth joining of curve segments with shape parameters and the construction of new curves have gradually attracted the attention of scholars at home and abroad,and its application in curve and surface modeling design plays an important role.Based on the Lupas q-Bezier curve,a new spline curve is obtained by considering the conditions to achieve G~2 smooth joining.When the parameters are given a certain value,the curve can be reduced to the Gamma spline curve and the B spline curve,respectively.In particular,for the cubic case,a set of parametric allocation functions is given,so that the generated curve can reach G~2continuity.The paper mainly includes the following parts:The first chapter is the introduction,which mainly introduces the background and significance of the research,summarizes the research status and basic knowledge of the q-analogue of Bernstein operator,and the Lupas q-Bezier curve at home and abroad.In the second chapter,the second derivative formula of Lupas q-Bezier curve is given and proved.Then an inverse symmetry of the Lupas q-Bezier curve is deduced.When the curve is controlled to be symmetrical,even if the shape parameters are changed,the curve itself is symmetrical,and its symmetrical"midpoint"is found.These conclusions supplement the properties of the Lupas q-Bezier curve,deepen the understanding of Lupas q-Bezier curve,and prepare for the follow-up chapter.In Chapter 3,the conditions for two adjacent n-order Lupas q-Bezier curve segments to achieve G~2 smooth joining are introduced in detail,and a new curve is obtained,which is called the Lupas q-G~2 spline curve(G~2 spline curve for abbreviation)in this paper.An example of spline curve design is analyzed,and the influence of two groups of shape parameters on the curve is observed.In particular,when special shape parameters are selected,the curve can degenerate into the classical Gamma spline curve.Theoretical analysis and calculation examples show that the G~2 spline curve has more flexibility in shape control than the Gamma spline curve.Chapter 4 defines a new cubic allocation function,obtains some properties of the function and its proof,and draws the graph of the function.Then a new curve is constructed by using this function,which is called cubic uniform Lupas q-B spline curve.The end-point property of the curve and its Lupas q-Bezier expression are obtained.Finally,the curve is extended to the surface,and the formation of the surface with shape parameters is analyzed.The fifth chapter summarizes and summarizes this paper,and puts forward the prospect for the next stage of research.