  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.
