Interactive Loop subdivision spline surface.
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(hold shift key or use mouse wheel to zoom it).
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We need N = 6 subdivisions to get smooth light reflex.
Loop subdivision spline surface
We use the modified Loop subdivision scheme . It is generalization of the
Lane-Riesenfeld algorithm for spline curves
- we subdivide every edge in two parts and make from every old triangle
four new small triangles
- average all new points with respect to the averaging mask
Po = αPo +
β ∑ k=1,n Pk .
- repeat 1-2 steps N times
- use evaluation mask to calculate the limit points
- use tangent masks to calculate normals (shall we use subdivided or limit points here?)
Modified averaging mask for a point with n neighbors is 
αnmod = 2 αn - 1,
βn = (1 - αn )/n,
where αn is the Loop averaging mask 
αn = [3/8 + cos(2π/n)/4]2 + 3/8.
Evaluation mask is
εn = 3/(11 - 8αn ).
Two tangents and tangent masks are
∑ k=1,n τn,k(j) Pk ,
τn,k(1) = cos(2πk/n),
τn,k(2) = sin(2πk/n)
and the normal direction is N = T1×T2 .
Similar to the Bezier spline patch, we can add per vertex materials and texture UVs as additional
dimentions and get colored surfaces (see models below).
- The Loop spline normals are not smooth at extraordinary points with 3 neighbor.
See also . It is not very important for models.
"Octahedron" is almost perfect.
- The script works only with closed, well oriented triangular meshes (without borders).
- The Loop algorithm takes ~200 lines of codes
(see loop.js public domain script).
Loop splines in 3D modelers
3D modeler Art of Illusion supports
approximating Loop subdivision splines.
Set Edit/Preferences/General/Interactive Surface Error = 0.01 and
Approximating Smoothing Method to play with Loop splines.
I used it to make the WebGL little dolphin.
The AoI model has only 100 vertices.
I exported it as Wavefront OBJ file (uncheck Subdivide Smooth Meshes before export).
Then converted points and faces from the OBJ
python convert_obj_js.py -i infile.obj -o outfile.js
Blender supports Catmull-Clark splines and Mikael Ohman wrote
subdivision surface modifier WIP draft.
Loop spline based models
Models from the Rome project.
In these models colors are blurry a bit due to per face material conversion into per vertex
in the scripts (and from textures into per face colors earlier as I think).
See further animated splines.
 Jos Stam & Charles Loop.
EUROGRAPHICS Computer Graphics Forum. November 2003, 22(1). pp. 79-85.
 Charles Loop Smooth Subdivision Surfaces Based on Triangles
M.S. Mathematics Thesis, University of Utah. 1987.
 Hartmut Prautzsch, Georg Umlauf
Improved Triangular Subdivision Schemes. Computer Graphics International 1998: 626-632
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updated 18 Apr 2013