| 三参曲线光滑融合的构造及参数的优化 |
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| 作者姓名： | 查东东 张 迪 刘华勇 |
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| 作者单位： | (安徽建筑大学数理学院，安徽 合肥 230601) |
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| 基金项目： | 安徽省自然科学基金项目(KJ2018A0518)；2020年度校级质量工程项目(2020kj15，2020jy22) |
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| 摘 要： | 在构造出含有多形状参数的三参基函数之后，分析了三参基函数的优良性质，又
定义了三参曲线，并给出了三参曲线的性质。再利用 G2 连续的定义，反求出三参曲线光滑融
合的条件，并组合了分段的三参曲线，且保证了组合曲线的连续性。还将三参曲线和原基函数
曲线、三次 B 样条曲线进行了多种性质的比较，看出三参曲线更贴近控制多边形，且满足 G2
连续的条件更简单，同时还具有调节曲线形状的优点。可以看到，通过改变不同参数取值构造
了各异的曲线旋转面模型。最后利用能量最小化模型，确定了三参曲线单一参数的取值，使三
参曲线过渡尽可能平滑。在此基础上，又研究了在弯曲能量约束下的组合三参曲线的参数优化，
并用曲率向量加以证明。最终得到唯一确定的具有能量约束的组合三参曲线。
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| 关 键 词： | 三参曲线 融合 连续性 能量优化 参数选取 |
Construction of smooth blending of three-parameter curves and optimization of parameters |
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| Authors: | ZHA Dong-dong ZHANG Di LIU Hua-yong |
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| Affiliation: | (School of Mathematics and Science, Anhui Jianzhu University, Hefei Anhui 230601, China) |
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| Abstract: | After constructing the three-parameter basis function with multiple shape parameters, this
paper analyzed the excellent properties of the three-parameter basis function, defined the
three-parameter curve, and presented the properties of the three-parameter curve. Next, based on the
definition of continuity, the condition for smooth blending of three-parameter curves was obtained,
and the segmented three-parameter curves were combed ensure the continuity of the combined curves.
Then, by comparing the properties of three-parameter curve with the original basis function curve and
the cubic B-spline curve, it can be seen that the three-parameter curve is closer to the control polygon,
easier to meet the continuous condition, and also capable of adjusting the curve shape. Different
models of curve rotating surface can be constructed by changing the values of different parameters. In
addition, the energy minimization model was used to determine the single parameter value of the
three-parameter curve, for the purpose that a most smooth transition of the curve can be accomplished.
On this basis, the parameter optimization of the combined three-parameter curve under the constraint of bending energy was also studied and proved by the curvature vector. Finally, a unique combined
three-parameter curve with energy constraints obtained. |
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| Keywords: | three-parameter curve blending continuity energy optimization parameter selection |
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