Re: Computation of limit positions for triangle/quad by spliting and averaging (with facet traversal)

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: : : I am wondering whether or not the limit position can be obtained in a similar manner that only requires facet traversal? For example, for the recent extension work you have done to quad/triangle subdivision scheme that improves continuity. It would be good if the same algorithm with different coefficients can be used. Otherwise the explict topological query for neighbors may be needed anyway for a subdivision system? : : The answer to your question is yes. For Catmull-Clark subdivision you simply accumulate a mask for each quad that looks like : : {{n/(n+5),2/(n+5)},{2/(n+5),1/(n+5)}}. : : Then you divide by the valence of the vertex. A similar mask holds for Loop subdivision. The triangle-quad scheme is a little more difficult. I haven't looked into whether or not there is a simple expression for the limit mask via this averaging approach. I suspect that a similar averaging mask holds though. : : One disadvantage to this technique is that other masks, such as the tangent masks, are not easy to compute via this averaging approach. So if your programs needs precise tangents, you should take this shortcoming into consideration. : : : My second question is a more general one. While it is true that subdivision surface can be defined by an arbitrary topological mesh, this can bring some "unwelcome" or inconvenient consequences. For example, the connectivity between the vertices can severely affect the final surface shape - this is unlike traditional NURBS where its surface geometry can be defined by control point geometry only (because its mesh topology is fixed). While this topological dependence may be considered as a consequence of admitting additional flexibility, would it imply that any editor system would need to allow the user change the connectivity as well as vertex geometry? What is your opinion here? : : I'm not completely sure I understand this question. NURBS systems really only operate on quadrilateral domains. You can restriction subdivision surfaces (namely Catmull-Clark) to the same types of domains that you use with NURBS. However, subdivision surfaces are more general in that we don't have to enforce smoothness constraints at extraordinary vertices and can handle arbitrary n-gons as well. : : In terms of an editing system, I would probably allow the standard polygon editing operations. The users should not only be able to manipulate the vertices of the shape, but also change the topology of the shape (add/delete polygons). It's true that the valence of a vertex strongly influences what the shape of the resulting surface is. However, I would be reluctant to limit what the user can do with the system. : : I'm thinking of a fully general modeling tool such as Maya. If your program's goal is something different, then these details may change depending on what you're trying to accomplish. Besides, these are only my opinions. The right people to ask are artists that already use similar types of software packages and determine how they create their models. Unfortunately, we tend not to have much access to the people trying to use the tools that we build. : ----- : Thanks for the comments. You are definitely right that limit position/normal masks become quite complicated and the quad/triangle paper from stam/loop actually mentioned that they compute the left eigenvectors numerically for irregular configurations. Also I have not seen any work related to boundary/crease treatment for this scheme - assuming the treatment will be an extension to Zorin's treatment for loop/catmullClark the scheme will become even more complicated. : Quite honestly from an application point of view, I am not sure if quad/triangle scheme competes well with a system with both loop and catmullClark schemes that are targeted at shapes of different characteristics.

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