二次双曲Bézier曲线曲面

为了简化构造组合曲线时,相邻曲线的控制顶点间应满足的光滑拼接条件,构造了一种结构类似于二次Bézier曲线的含参数的双曲型曲线,称之为H-Bézier曲线。该曲线具有Bézier曲线的许多基本性质,如凸包性、对称性、几何不变性、端点插值和端边相切性。另外,该曲线具备形状可调性,可以精确表示双曲线。此外,若取特殊的参数,则当相邻H-Bézier曲线的控制顶点间满足普通Bézier曲线的G1光滑拼接条件时,曲线在公共连接点处可以达到G3光滑拼接。另外,给出了构造与给定多边形相切的H-Bézier曲线的方法,该方法简单有效,而且整条曲线对给定的切线多边形是保形的。运用张量积方法,将H-Bézier曲线推广后得到的曲面同样具有很多良好的性质。

Quadratic hyperbolic Bézier curve and surface

When composite curves are constructed, the control points of the adjacent curves must meet certain conditions. In order to simplify the conditions, a kind of hyperbolic polynomial curve with parameters, called H-Bézier curve, which has a structure similar to the quadratic Bézier curve, is constructed. This curve has many basic properties of Bézier curve, such as convex hull property, symmetry, geometric invariance, endpoint interpolation and end edge tangent property. Besides, the curve has shape adjustability and can represent hyperbola. If special parameters are chosen, then when the control points of two adjacent curves satisfy the  conditions for Bézier curve, they can achieve   continuity. In addition, the method of constructing H-Bézier with given tangent polygon is given. This method is simple and effective, and the entire curve is conformal to the tangent polygon. Using tensor product method, the H-Bézier curve can be extended to surface, which also has many good properties.