Catmull-Clark subdivision surfaces: corners and creases

Posted: 1306586508|%e %B %Y, %H:%M|agohover
Tags: catmull clark subdivision surface

This post delves a bit further into the realm of subdivision surfaces. Building upon the lore gathered in the previous post, it describes extensions to the Catmull-Clark scheme for modeling fillets and blends.

# Sharp features

Because smooth only surfaces are of limited use, Pixar people have introduced sharp features into Catmull-Clark subdivision surfaces: corners and creases. A few more rules (let's call them sharp rules) have been added to the smooth subdivision rules.

If you remember my previous post, the subdivision produces three types of vertices:

• face vertex. The vertex uses always the smooth rule
• edge vertex. For each edge tagged as sharp, the new vertex is the average of the edge's endpoints. The new sub-edges are also tagged as sharp. The other edges use the smooth rule.
• control vertex. For each control vertex of the mesh, the vertex is moved to a new location that depends on the number of sharp edges incident at the vertex.
• If this number is less than 2, the vertex uses the smooth rule.
• If the number equals 2, the vertex uses the crease vertex rule: the new vertex is a weighted average of the old vertex location (3/4) and of the two other endpoints of the incident creases (1/8).
• If the number is more than 2, the vertex uses the corner rule: the vertex does not move under subdivision.
• Of course, if a vertex is tagged as sharp, it uses the corner rule whatever is the number of incident sharp edges.

Here is a small example of how a surface changes, just by tagging edges or vertices. The control mesh is the same for each picture.

 Smooth surface wireframe control mesh, where smooth edges are yellow
 Creases creases edges are red
 Corner the upper right vertex is a corner

In this example, four edges meet at the same vertex, implicitely making a corner out of it.

 Implicit corner # Semi-sharp features

In the real world, surfaces are never infinitely sharp. Anything when viewed sufficiently closely is smooth. Another refinement of the subdivision rules is needed to obtain semi-sharp features. For that purpose, sharp edges and corners are weighted by a sharpness parameter. This parameter is decreased by 1 at each iteration of the subdivision process and controls which rules are applied to sharp features.
* If the sharpness is greater or equal to 1, sharp subdivsion rules are applied
* If the sharpness is between 0 and 1, the result is a linear interpolation of the smooth rule and of the sharp rule using the sharpness value.
* If the sharpness is lower or equal to 0, smooth subdivision rules are applied.

This is a very intuitive mechanism which behaves like levels of detail. At coarser levels of the subdivision, features are sharp and become smooth at finer levels.

Here is how the smooth surface of the first paragraph is affected by various sharpness values. sharpness = 1.0 sharpness = 2.0 sharpness = 3.0

In the following animation, the sharpness varies between 0 and 10 with a 0.1 step.

Notice that a crease does not need to be not a closed contour. In the next example, only three of the four upper edges are tagged as creases with a sharpness of 4.0.

 Open crease # More to come

We are not yet done with subdivision surfaces. The next post will describe how to deal with holes in the control mesh and boundaries for open meshes.

Bibliography
1. Tony Derose, Michael Kass, and Tien Truong. "Subdivision surfaces in character animation." In Proceedings of the SIGGRAPH 1999 annual conference on Computer graphics, pages 85-94, Los Angeles, CA, USA,August 1999.

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