Construction Of Bézier Curves By Energy Constrains

Computer-aided geometric design(CAGD for short),mainly studies the theory,algorithm and application of geometric modeling.And the representation and approximation of curves and surfaces is one of the most important contents of CAGD research.Because of its good properties,Bézier method has become a kind of important tool in curves and surfaces modeling.CAGD has now become a cross-discipline involving multiple fields,and has a very important position in modern industry and even architectural design,clothing models,and robotics with the continuous development of CAGD theory and application.In order to meet the needs of designers well,CAGD usually requires certain constrains in the process of constructing curves and surfaces.Among all the constrains,energy constrains are widely used because of its physical and geometric meanings.This paper gives a method of constructing Bézier curves with energy constraints.For the Bézier curve with given first and last control vertices and initial tangent vector,the remaining control vertices can be uniquely determined by the given conditions and an unknown parameter a with the constraint of Jerk energy.The parameter a represents the length of the initial tangent vector and can be uniquely determined by bending energy constraint on the curve,so that a single Bézier curve or a combined Bézier curve withG~1 continuity can be explicitly constructed by the constraints of Jerk energy and bending energy.Furthermore,this paper presents an algorithm for constructing open or closed combined Bézier curve withG~1continuity that satisfies the above energy constraints on an ordered set of points.The feasibility and effectiveness of the algorithm in this paper are verified by several examples.The algorithm also has certain advantages in terms of energy value and algorithm time when compared with other methods.The paper is structured as follows:Chapter 1 introduces the research background of this article;Chapter 2 introduces the definition,nature,algorithm and energy estimation formula of Bézier curve;Chapter 3 gives algorithm of constructing single or combined Bézier curves with constraints of Jerk energy and bending energy;Chapter 4 verifies the validity of the algorithm through several typical examples when compared with other methods.