# To what plane curve does a Catmull-Clark subdivision converge?

1 BU ×1 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?)

• Just to note, the same is true with a cube. – gandalf3 Sep 11 '13 at 23:54
• @gandalf3 Yeah, but I can't put 10 levels of subsurf on a cube without my computer freezing. – wchargin Sep 12 '13 at 13:26
• I tested it, and it does work. (though it's quite slow..) – gandalf3 Sep 12 '13 at 19:29
• @gandalf3 I'm sure it works, but my computer can't handle it... it overheats and shuts down :\ – wchargin Sep 12 '13 at 23:41

## 2 Answers

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.

• Thanks for your answer, that's a useful link and a helpful quote; however, it doesn't really answer what it does converge to. – wchargin Sep 12 '13 at 13:28
• @WChargin As far as I understand: It doesn't converge to a particular shape it only depends on the control mesh and the B-splines. There is another paper on "Behaviour of recursive division surfaces" cs.berkeley.edu/~sequin/CS284/PAPERS/DooSabin_SDSurf.pdf – stacker Sep 12 '13 at 14:12
• I understand the nature of the question. If it was about bezier curves, the answer would be that it converges to parts of parabolic curves. Sadly I don't have a clue about the case with nurbs. Maybe a question in the mathematics section of SE will be helpful. – Haunt_House Sep 12 '13 at 15:19

The limit of a Catmull-Clark surface is the union of (infinitely many) uniform cubic B-spline patches. The "infinitely many" bit is in parentheses because that's only true when the surface contains singularities (that come from non-4-valent, "extraordinary" vertices).

For the curve case, it's much simpler: you just get the uniform cubic B-spline that is defined by the control points you started with. Spline means "piecewise polynomial", so this means the limit is a collection of parametric polynomials, joined together (with continuity of curvature) at positions called knots.

One way of seeing that this is true is that each subdivision step is actually performing knot insertion on the original curve, splitting each cubic piece into two and (roughly) doubling the number of knots in the spline.