Cubic G~3 Spline Curve

Curve modeling has a long history in computer graphics.The basic goal of curve modeling is to construct curves which have possessed good geometric properties and can be edited easily.In industrial or conceptual design,one important question is how to construct a fair freeform curve.Non-Uniform Rational B-Spline(NURBS)have become the industry standard in the modeling of freeform curve and surface for the powerful representation and the facile control method.Cubic spline is the primarily choice for the industrial applications cause the low computation and the enough smoothness in most aspects.So,to construct more smooth cubic spline has become a subject worth studying.Nowadays,the cubic spline can be C2(or G2)continuous at most,which result in the curvature of curve is only continuous rather than smooth.This paper proposes a method to construct a G3 cubic spline curve from any given control polygon.Given a control polygon consist of a ordered points Pi,we can se-lect two points B3(i-1)+1 and B3(i-1)+2 called inner Bezier points on each edge PiPi+1,Then we can determine a Bezier junction point B3i according the G2 constraint and the inner Bezier points lying on two adjacent edges.A cubic Bezier segment can be de-fined by the two inner Bezier points B3(i-1)+1,B3(i-1)+2 and two Bezier junction points B3(i-1),B3i.The G2 cubic spline curve can be formed by combining all these segments.This cubic spline curve are G2 continuous at each Bezier junction point B3i.Accord-ing to the G3 constraints,we solve a non-linear system to obtain suitable inner Bezier points such that the cubic spline curve is G3 continuous.We prove that the solution of the system always exists under a restriction on the control polygon.Our G3 cubic spline curve can be easily edited and modified by editing and modifying the control polygon similarly as B-splines.And the numerical examples show that the curvature combs and the curvature plots of the G3 cubic spline curve are more smooth in contrast with the traditional C2 cubic spline curve.