With the rapid development of science and technology

geometric modeling techniques for curves have recently gained significant research interest. According to the 2D and 3D freeform deformation of parametric curves and free curves

a new algorithm based on polynomial extension function with platform is proposedto obtain several deformation effects. The new deformation method possesses perfect properties such as interval peak

peak value and symmetries. Furthermore

it has the closure property for deformations of Bézier and NURBS curves which are the mainFstream curves of present modeling system. A simple extension function is first presented. Unlike most existing freeform deformation methods

the proposed extension function has an ordinary polynomial form. Hence

this extension function does not include transcendental functions and complex functions. With this extension function

one new extension factor with extension and smoothing parameters is also established. This new extension factor constructed with the extension function possesses perfect properties

including interval peaks

peak values

and symmetries. Finally

the extension matrixes are applied to the parametric and freeform curves to obtain the global

local

periodic

and elastic deformation effects. This objective is achieved by adjusting the deformation intervals

extension parameters

smoothing parameters

and deformation directions. The deformation method remains self-enclosed when implemented on Bezier and NURBS curves

which are mainstream curves in CAGD. Several numerical examples show that the proposed method is easy to calculate and control and that it can be used repeatedly to achieve a large number of contour lines. Compared with many freeform deformation methods

the proposed algorithm can be used for not only general 2D and 3D parametric curves but also freeform curves. Thus

it demonstrates extensive application scope. Because the extension function has perfect properties such as interval peak

peak value and symmetries

it can produce different kinds of special curves with angular points and cuspidal points which the other deformation method can't achieve. In sum

the proposed algorithm greatly enriches the deformation effects of curves.