To what plane curve does a Catmull-Clark subdivision converge? 1&thinsp;BU &times;1&thinsp;BU plane, 10 levels of Subdivision Surface modifier set to Catmull-Clark:



That's not a circle.

What is it?

(Mathematically, what's the limit surface of a Catmull-Clark subdivision?)
According to the original paper CATMULL, E., AND CLARK, J. Recursively generated B-spline surfaces on arbitrary topological
meshes. Computer-Aided Design 10 (Sept. 1978), 350–355.

For rectangular control-point meshes the Catmull Clark subdivision modifier creates a standard B-spline surface. Regardless of the number of subdivision it will not converge to an accurate circle, but with more control points the difference between a circle and the interpolated shape will be less noticeable.