中文摘要：

针对参数曲面的自由变形，构造了矩形域上一类带平台的、分片多项式形式的伸缩函数。新的伸缩函数具备了对称性、单点峰值性、线性峰值性和区域峰值性等良好的性质；且具备了简单的多项式形式，易于构造和操控；由此构造的伸缩因子中的各个参数具备明显的几何意义，特别适用于交互设计。将基于伸缩因子构造的变形矩阵作用于待变形的曲面，可以获得诸如凹凸、剪切、伸缩和改变变形中心等各类变形效果，借助于叠加变形可进一步实现丰富多样的复杂变形效果。该文付出了大量的数值实例，展示了各类变形效果以及伸缩函数中各个不同参数对形状的调控。

英文摘要：

According to free form deformation of parametric surfaces , a new method based on extension function is proposed .It is made by piecewise polynomials and defined on rectangular region . Based on these ,the new extension factor we constructed not only possesses perfect properties such as symmetry ,single peak ,linear peak and region peak , but also holds some parameters which have obvious geometric meanings .In real applications ,the extension factor is very suitable for interactive design due to these properties and extraordinary parameters .Furthermore ,the new extension factor is easy to construct and convenient to control because of its simple polynomial forms . Applying the deformation matrix constructed by the new extension factor to the original surfaces’ equations , plentiful deformation surfaces can be achieved . Deformation effects such as shearing , concave ＆convex ,expand ＆ contract and changing of deformation center can be obtained .With the help of superimposing ,more complex deformation effects can be realized .Lots of numerical experiments are given at the end of paper which illustrate different kinds of deformation effects and plentiful adjusting effects of different parameters .