Subdivision surfaces are considered as an extension of splines to accommodate models with complex topologies, making them useful for addressing PDEs on models with complex topologies in isogeometric analysis. This has generated a lot of interest in the field of subdivision space approximation. The quasi-interpolation offers a highly efficient approach for spline approximation, eliminating the necessity of solving large linear systems of equations. Nevertheless, the lack of analytical expressions at extraordinary points on subdivision surfaces makes traditional techniques for creating B-spline quasi-interpolants inappropriate for subdivision spaces. To address this obstacle, this paper innovatively reframes the evaluation issue associated with subdivision surfaces as a correlation between subdivision matrices and limit points, offering a thorough method for quasi-interpolation specifically designed for subdivision surfaces. This developed quasi-interpolant, termed the subdivision space projection operator, accurately reproduces the subdivision space. We provide explicit quasi-interpolation formulas for various typical subdivision schemes. Numerical experiments demonstrate that the quasi-interpolants for Catmull-Clark and Loop subdivision exhibit third-order approximation in the (L_2) norm and second-order in the (L_\infty) norm. Furthermore, the modified Loop subdivision quasi-interpolant achieves optimal approximation rates in both the (L_2) and (L_\infty) norms.

翻译：细分曲面被视为样条方法在复杂拓扑模型上的延伸，使其在等几何分析中能有效处理复杂拓扑模型上的偏微分方程问题，这引发了细分空间近似领域的研究热潮。准插值方法为样条逼近提供了一种高效途径，无需求解大型线性方程组。然而，由于细分曲面奇异点处缺乏解析表达式，传统B样条准插值构造方法无法直接应用于细分空间。针对这一难题，本文创新性地将细分曲面求值问题重构为细分矩阵与极限点的关联关系，提出了一套专用于细分曲面的准插值构造理论。所构建的准插值算子（称为细分空间投影算子）具有细分空间精确再现性。我们给出了多种典型细分格式的显式准插值公式。数值实验表明，Catmull-Clark细分和Loop细分的准插值在(L_2)范数下具有三阶逼近精度，在(L_\infty)范数下具有二阶逼近精度；而改进的Loop细分准插值在(L_2)与(L_\infty)两种范数下均达到最优逼近阶。