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Re: subdive a square

Posted by hjb_zy on April 05, 2005 at 15:01:51:

In Reply to: Re: subdive a square posted by Alex on March 26, 2005 at 23:36:33:

: : : I am a newbie in subdivision. I am wondering some issues on subdividing a square (surface) by Catmull-Clark scheme. For the first step, the boundary rules need to be used for generating new edge points and vertex points since all four edges are boundary edges. By applying the boundary rules, finally, the surface boundary will become a cubic bspline curve.

: : : My question is:
: : : Is the final surface a standard bi-cubic B-spline surface?

: : : If yes, can we say the bi-cubic B-spline surface is only controlled by the four origin square vertices, and what will be the mathematical expression like?

: : : If no, does it mean that it’s composed by some standard bspline patches and surrounding boundary patches? If so, is there any special for the boundary patches and how can we exactly evaluate these patches as did in J.Stam’s paper? Or could you point out some references for discussion on the boundary issues.

: : The answer to your question is no. The surface is not a standard bicubic patch. It is not composed of some standard b-spline patches either. Well, that's not quite true. Similar to an extraordinary vertex in Catmull Clark subdivision, there is an infinite set of regular bicubic patches that constantly decrease in size as you approach the boundary. This is always the case with subdivision rules that are only altered in a finite area of the mesh though.

: Thanks, Scott.
: My further question is how to analyze the surface continuity around the boundary, and if there is a possible way to exactly evaluate the surface near boundary without subdividing it thoroughly. Or is there any reference you can point out?

My understanding is that subdividing a square and treating all four edges as boundaries does not produce a patch - it just produces four cubic spline curve segments, nothing more. This scenario seems to be a degenerated case of how the boundary rule can affect the surface definition near the boundary - in this case the surface definition is overwhelmed by the boundary definition such that the surface is not well defined in between. For example, if the square is repeatly subdivided the boundary curve will get refined but no information can be provided for the interior geometry.

Manuscript "Evaluation of Piecewise Smooth Subdivision Surfaces" by Zorin at http://mrl.nyu.edu/publications/loop-evaluation/ may be of help for your purpose.

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: : : : I am a newbie in subdivision. I am wondering some issues on subdividing a square (surface) by Catmull-Clark scheme. For the first step, the boundary rules need to be used for generating new edge points and vertex points since all four edges are boundary edges. By applying the boundary rules, finally, the surface boundary will become a cubic bspline curve. : : : : My question is: : : : : Is the final surface a standard bi-cubic B-spline surface? : : : : If yes, can we say the bi-cubic B-spline surface is only controlled by the four origin square vertices, and what will be the mathematical expression like? : : : : If no, does it mean that it’s composed by some standard bspline patches and surrounding boundary patches? If so, is there any special for the boundary patches and how can we exactly evaluate these patches as did in J.Stam’s paper? Or could you point out some references for discussion on the boundary issues. : : : The answer to your question is no. The surface is not a standard bicubic patch. It is not composed of some standard b-spline patches either. Well, that's not quite true. Similar to an extraordinary vertex in Catmull Clark subdivision, there is an infinite set of regular bicubic patches that constantly decrease in size as you approach the boundary. This is always the case with subdivision rules that are only altered in a finite area of the mesh though. : : Thanks, Scott. : : My further question is how to analyze the surface continuity around the boundary, and if there is a possible way to exactly evaluate the surface near boundary without subdividing it thoroughly. Or is there any reference you can point out? : My understanding is that subdividing a square and treating all four edges as boundaries does not produce a patch - it just produces four cubic spline curve segments, nothing more. This scenario seems to be a degenerated case of how the boundary rule can affect the surface definition near the boundary - in this case the surface definition is overwhelmed by the boundary definition such that the surface is not well defined in between. For example, if the square is repeatly subdivided the boundary curve will get refined but no information can be provided for the interior geometry. : Manuscript "Evaluation of Piecewise Smooth Subdivision Surfaces" by Zorin at http://mrl.nyu.edu/publications/loop-evaluation/ may be of help for your purpose.

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