Research On Cubic Pythagorean-hodograph Curve

Offset curves,also known as parallel curves,have become the focus of Computer Aided Geometric Design(CAGD)research.Pythagorean-hodograph(PH)curves are widely used in curve modeling because they can accurately calculate the arc length of curves and accurately represent isometric curves in rational form.For some discrete data interpolation,PH curve can not only produce smoother curve trajectory than classical parametric polynomial curve,but also calculate the curve bending energy more accurately.At the same time,the PH curve remains exactly consistent with the standard Bézier representation.In this paper,we mainly study the construction of two cubic PH curves with G~2continuous and specified total length at the splicing point,and extend the concept of PH curve to hyperbolic space,define plane cubic PH-H curve and construct cubic PH-H curve satisfying G~1interpolation.First,this paper briefly summarizes the development history of CAGD,the PH curve and its wide application in CAGD.The definition,properties and research status of PH curve are introduced in detail,and the algorithm of Hermite interpolation problem is given.And a brief introduction to Algebraic hyperbolic Bézier curve.Then,by taking advantage of the fact that the arc length of the PH curve can be expressed by an exact polynomial,This paper discusses the G~2continuous blending of cubic PH curves under total arc length constraint.Given three points including two end control points and a joint point,construct two cubic PH curves such that they interpolate the end control points and are G~2continuous at joint point with prescribed total arc length.It can also be regarded as a curve extension problem.According to the arc length formula of cubic PH curve and the condition of G~2blending,the mathematical model is established,and then the model is solved.After comparison of algorithms,the problem is transformed into a constraint constrained minimization problem,which can solve the problem quickly and directly.Several examples serve to illustrate our method.Finally,according to the good geometric properties of AH Bézier curves,the algebraic space is extended,the concept of PH curve from univariate polynomial space is generalized to algebraic hyperbolic space,proposed constructs three PH-H curve method.The cubic PH-H curve of a plane is defined and its properties are given.Specifically,by using hyperbolic basis functions and algebraic Bézier basis functions respectively,two different necessary and sufficient conditions for a planar cubic algebraic hyperbolic Bézier curve to be a PH curve are obtained.Moreover,cubic PH-H curves are applied in the problem of G~1Hermite interpolation with determined closed form solutions.There are three cases of the existence of solutions,and different examples are given to illustrate these three cases.