自动实现G^(1)连续的组合曲线曲面构造方法

Automatic Realization of G^(1) Continuous Combined Curve Surface Construction Method

G^(1)连续是一个合理且较为简便的光滑性选择,虽然Bézier曲线曲面实现G^(1)连续对于控制顶点的位置要求相对较低,但控制顶点的选取仍然需要一定的计算,且Bézier曲线曲面缺乏独立于控制顶点的形状表示自由度。为了使曲线曲面的G^(1)拼接条件更加简单,同时增加曲线曲面形状表示的自由度,构造了一系列含参数的多项式基函数,并由之定义了结构类似于n(n≥2)次Bézier曲线曲面的新曲线曲面。其保留了Bézier方法的诸多性质,其中的二次、三次曲线端点位置可调,更高次数的曲线端点位置和内部形状皆可调。值得一提的是,在拼接时,不管曲线的次数是多少,只要前一段的最后一条控制边与后一段的第一条控制边重合,且二者的参数之间满足一定的关系,二者即可实现G^(1)连续。这样一来,任给一条控制多边形,无需做任何预处理,即可方便地构造G^(1)连续的组合曲线,曲线的各部分可以由不同数量的控制顶点定义,对于曲面也有类似的结论。

G^(1) continuity is a reasonable and relatively simple choice for smoothness,although Bézier curves and surfaces to achieve G^(1)continuous for the position of the control point is relatively low.However,the selection of control point still needs some calculation,and Bézier curves and surfaces lacks independence and control point shape to represent the degree of freedom.In order to simplify the conditions of G^(1) splicing of curves and surfaces,and increase the degree of freedom represented by curve and surface shape,a series of polynomial basis functions with parameters and constructed,a new curve and surface is similar to that ofn(n≥2)order Bézier curve and surface is defined.They retain many properties of the Bézier method,and the position of the quadratic and cubic curve endpoints can be adjusted,and the position of the higher number of curve endpoints and the internal shape can be adjusted.It is worth mentioning that in the splicing,no matter what the number of curves is,as long as the last control edge of the previous section coincides with the first control edge of the following section and the parameters of the two meet a certain relationship,the G^(1) continuity of the two can be realized.In this way,given any control polygon,G^(1) continuous composite curves can be easily constructed without any preprocessing.Each part of the curve can be defined by a different number of control points.Similar results can be obtained for surfaces.