We present an approximation problem of surfaces of a generalized offset surface with offset variable distances and directions. Such approximating surface fits some given data points and minimizes a Sobolev’s semi-norm of order 3. The study of the new results, from a mathematical point of view, carefully establishing the proof of the convergence between the generalized offset surface and its approximating spline in an adequate parametric bicubic spline space. Moreover, the approximating spline function is computed and an estimation of the relative error is introduced. Finally, some numerical and graphic examples are shown in order to prove the useful and the effectiveness of our method.

中文翻译：

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通过双三次样条近似广义偏移曲面

我们提出了具有偏移距离和方向可变的广义偏移曲面的曲面的近似问题。这种近似曲面拟合了一些给定的数据点并最小化了 3 阶 Sobolev 的半范数。 新结果的研究，从数学的角度，仔细地建立了广义偏移曲面与其近似样条之间收敛的证明一个足够的参数化双三次样条空间。此外，计算近似样条函数并引入相对误差的估计。最后，给出了一些数值和图形示例，以证明我们方法的有用性和有效性。