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Subdivision surface 202
used a four-directional box spline to build the scheme. This scheme generates C 1 continuous limit surfaces on
initial meshes with arbitrary topology.
• √3 subdivision scheme - This scheme has been developed by Kobbelt (2000) and offers several interesting
features: it handles arbitrary triangular meshes, it is C 2 continuous everywhere except at extraordinary vertices
where it is C 1 continuous and it offers a natural adaptive refinement when required. It exhibits at least two
specificities: it is a Dual scheme for triangle meshes and it has a slower refinement rate than primal ones.
Interpolating schemes
After subdivision, the control points of the original mesh and the new generated control points are interpolated on the
limit surface. The earliest work was the butterfly scheme by Dyn, Levin and Gregory (1990), who extended the
four-point interpolatory subdivision scheme for curves to a subdivision scheme for surface. Zorin, Schröder and
Swelden (1996) noticed that the butterfly scheme cannot generate smooth surfaces for irregular triangle meshes and
thus modified this scheme. Kobbelt (1996) further generalized the four-point interpolatory subdivision scheme for
curves to the tensor product subdivision scheme for surfaces.
• Butterfly, Triangles - named after the scheme's shape
• Midedge, Quads
• Kobbelt, Quads - a variational subdivision method that tries to overcome uniform subdivision drawbacks
Editing a subdivision surface
Subdivision surfaces can be naturally edited at different levels of subdivision. Starting with basic shapes you can use
binary operators to create the correct topology. Then edit the coarse mesh to create the basic shape, then edit the
offsets for the next subdivision step, then repeat this at finer and finer levels. You can always see how your edit
effect the limit surface via GPU evaluation of the surface.
A surface designer may also start with a scanned in object or one created from a NURBS surface. The same basic
optimization algorithms are used to create a coarse base mesh with the correct topology and then add details at each
level so that the object may be edited at different levels. These types of surfaces may be difficult to work with
because the base mesh does not have control points in the locations that a human designer would place them. With a
scanned object this surface is easier to work with than a raw triangle mesh, but a NURBS object probably had well
laid out control points which behave less intuitively after the conversion than before.
Key developments
• 1978: Subdivision surfaces were discovered simultaneously by Edwin Catmull and Jim Clark (see Catmull–Clark
subdivision surface). In the same year, Daniel Doo and Malcom Sabin published a paper building on this work
(see Doo-Sabin subdivision surfaces.)
• 1995: Ulrich Reif solved subdivision surface behaviour near extraordinary vertices. [2]
• 1998: Jos Stam contributed a method for exact evaluation for Catmull–Clark and Loop subdivision surfaces under
arbitrary parameter values. [1]