高阶F连续的扩展Bézier样条

High-Order F-Continuous Extended Bézier Spline

摘要: 为实现曲线在$C^3\cap F^4$ 连续性约束下的光滑拼接，并能够在保持控制点不变的前提下灵活调整曲线形状，本文提出了一种结构类似于Bézier曲线的新型曲线。首先，构建了调配函数初始形式。随后，依据目标样条的性质，推导该调配函数在端点处应满足的特性，并据此建立方程组；通过求解该方程组，确定调配函数的具体形式。最终，将调配函数与控制点进行线性组合，定义出一种新的7次扩展Bézier曲线。该曲线不仅继承了Bézier曲线的凸包性、几何不变性和仿射不变性等基本性质，还在拼接过程中具备保持$C^3$ 连续的同时自动满足$F^4$ 连续的能力。因此，在不改变控制点、连续性条件的前提下，仍可灵活调控曲线形状。

Abstract: In order to achieve the smooth splicing of curves under the constraint of $C^3\cap F^4$ continuity and to flexibly adjust the shape of the curve while keeping the control points unchanged, this paper proposes a new type of curve with a structure similar to that of a Bézier curve. First, the initial form of the blending function is constructed. Subsequently, based on the properties of the target spline, the characteristics that the blending function should satisfy at the endpoints are derived, and an equation system is established accordingly. By solving this equation system, the specific form of the blending function is determined. Finally, the blending function is linearly combined with the control points to define a new type of 7th-order extended Bézier curve. This curve not only inherits the basic properties of Bézier curves, such as the convex hull property, geometric invariance, and affine invariance, but also has the ability to maintain $C^3$ continuity while automatically satisfying $F^4$ continuity during the splicing process. Therefore, it is still possible to flexibly adjust the shape of the curve without changing the control points and the continuity conditions.