We are interested in constructing more generalized barycentric coordinates (GBC) over arbitrary polygon in the 2D setting. We propose a constrained minimization over the class of infinitely differentiable functions subject to the GBC constraints of preserving linear functions and the non-negativity condition. It includes the harmonic GBC, biharmonic GBC, maximum entropy GBC, local barycentric coordinates as special cases. We mainly show that the constrained minimization has a unique solution when the minimizing functional is strictly convex. Next we use a \(C^r\) smoothness spline function space \(S^r_d(\triangle )\) with \(r\ge 2\) over a triangulation \(\triangle \) of a polygon of interest in \(\mathbb {R}^2\) to approximate the minimizer. One advantage of using smooth splines is that derivaties, e.g. the mean curvature and/or Gaussian curvature of spline GBC functions can be calculated. As the minimization restricted to the spline space \(S^r_d\) certainly has a unique minimizer, we use the standard projected gradient descent (PGD) method to approximate the spline minimizer. To find the projection of each iteration, we shall explain an alternating projection algorithm (APA). A convergence of the APA and the convergence of the PGD with the APA will be presented. As an example of this approach, a new kind of biharmonic GBC functions which preserve the nonnegativity is constructed. Finally, we have implemented the PGD method based on bivariate splines of arbitrary degree d and arbitrary smoothness r over arbitrary triangulation as long as \(d>>r\). The surfaces of many new GBC’s will be shown. Some standard GBC applications will be demonstrated.

我们对在2D设置中的任意多边形上构造更广义的重心坐标（GBC）感兴趣。我们在保留线性函数和非负性条件的GBC约束下，针对一类无限可微函数提出了约束最小化。作为特殊情况，它包括谐波GBC，双谐波GBC，最大熵GBC，局部重心坐标。我们主要表明，当最小化函数严格凸时，约束最小化具有唯一的解决方案。接下来，我们在感兴趣的多边形的三角剖分\（\ triangle \）上使用\（C ^ r \）平滑样条函数空间\（S ^ r_d（\ triangle）\）与\（r \ ge 2 \）\（\ mathbb {R} ^ 2 \）近似最小化器。使用平滑样条的一个优点是可以计算样条GBC函数的平均曲率和/或高斯曲率。由于限于样条空间\（S ^ r_d \）的最小化当然具有唯一的最小化器，因此我们使用标准投影梯度下降（PGD）方法来近似样条最小化器。为了找到每次迭代的投影，我们将解释一种交替投影算法（APA）。将介绍APA的收敛性以及PGD与APA的收敛性。作为该方法的示例，构建了一种保留非负性的新型双谐波GBC函数。最后，我们基于任意度为d的双变量样条实现了PGD方法只要\（d >> r \），就可以在任意三角剖分上获得任意平滑度r。将显示许多新GBC的表面。将演示一些标准的GBC应用程序。