Limit properties of rational Bézier curves with mixed weights(具有混合权函数的有理Bézier曲线的极限性质) - 魏琦文(WEI Qiwen); 刘向云(LIU Xiangyun); 吴金明(WU Jinming); 朱春钢(ZHU Chungang) (Zhejiang Daxue xuebao. Lixue ban)
The weights of a rational Bézier curve play a crucial role in shape modification of the curve. As the weight of a rational Bézier curve approaches infinity, the curve tends towards the corresponding control point. Previous research has shown that when all weights of a rational Bézier curve tend to infinity in the form of power or exponential functions, the curve approaches its regular control curve. In this paper, we propose a new model, combining the transformation relationship between power functions and exponential functions, and then define the regular control curve of a rational Bézier curve with mixed weight functions. Based the toric degeneration theory, we prove that the limit of a rational Bézier curve is exactly its regular control curve when all weights tend to infinity in the form of mixed functions.(权因子是调整有理Bézier曲线形状的重要手段。当单个权因子趋于无穷时，有理Bézier曲线趋向于相应的控制顶点。已有研究表明，当所有权因子都以幂函数或指数函数形式趋于无穷时，有理Bézier曲线的极限曲线为其正则控制曲线。基于此，提出了一个新模型，结合幂权函数与指数权函数的转换关系，定义了具有混合权函数的有理Bézier曲线的正则控制曲线；结合toric退化理论，证明了当所有权因子都以混合函数形式趋于无穷时，有理Bézier曲线的极限曲线恰为其正则控制曲线；最后通过实例验证了结论的正确性。)