Loop surfaces are similar to Doo-Sabin or Catmull-Clark surfaces in that they are based upon the subdivision paradigm. However, as the Doo-Sabin and Catmull-Clark methods are based upon quadratic and cubic uniform B-spline surface subdivision, the Loop algorithm is based upon the subdivision of quartic uniform box splines - and therefore a mesh of triangles.
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Given a triangular mesh, the Loop refinement scheme generates both vertex points and edge points and utilizes the following subdivision masks
The vertex mask generates new control points for each vertex, and the edge masks generate new control points for each edge of the original triangular mesh.
The edge masks compute the new edge points as the average of three
values : the two centers of the faces that share the edge and the
midpoint of the edge. The vertex mask can be stated as a convex
combination of the points , the original vertex, and
average of the original points that share an edge with . This
convex combination can be seen to be the following: If
new vertex point, then
Specifying the Refinement Procedure
For an arbitrary triangular mesh we can apply the same rules to generate the new edge points and new vertex points for the refined mesh. However, Loop noted that with the above vertex rule, the surface exhibited some points for which a tangent plane was discontinuous. Upon further examination, he noted that a rule of the form
These surfaces are known as Loop surfaces and are generated with the subdivision masks
where for a vertex of edges, is utilized to find the new vertex control point. That is