Subdivision

Surface subdivision is a geometric modeling technique that defines smooth curves or surfaces as the limit of a sequence of successive refinements.

Developed as a generalization of spline surfaces, subdivision allows for the representation of complex, arbitrary control meshes while avoiding the topological constraints and "cracking" issues often associated with traditional Non-Uniform Rational B-Splines (NURBS).

Various algorithms have been established to handle different mesh types and desired continuity:

• The Catmull-Clark scheme is frequently used for quadrilateral meshes to produce $G^2$ continuous surfaces, while
• The Loop subdivision scheme is a popular approximating method specifically designed for triangular meshes.

By iteratively applying simple refinement rules—typically involving a "splitting" step to increase resolution and an "averaging" step to relocate vertices—subdivision transforms a coarse initial shape into a highly detailed, smooth limit surface suitable for high-end animation and scalable rendering.

Octa Loop

This example uses Loop subdivision to transform an octahedron into a sphere. The results below are based on Octa-Loop Subdivision Scheme (GitHub).

We create a unit radius octahedron template, and successively refine it into a sphere. The sphere is a useful baseline subject of study because it:

• Can easily be approximated by a voxel stack at various resolutions,
• Can easily be approximated by a finite element mesh,
• Has a known analytic volume, and
• Has a known analytic local curvature.

Base Octahedron

We created a unit radius template, octa_base.obj, with contents listed below:

v 1.0 0.0 0.0
v 0.0 1.0 0.0
v -1.0 0.0 0.0
v 0.0 -1.0 0.0
v 0.0 0.0 1.0
v 0.0 0.0 -1.0
f 1 2 5
f 2 3 5
f 3 4 5
f 4 1 5
f 2 1 6
f 3 2 6
f 4 3 6
f 1 4 6

Refinement

The refinement below was created with MeshLab 2022.02, Subdivision Surfaces LS3 Loop, based on Boye et al.¹

name	image	file size
octa_loop0.stl		2.1kB
octa_loop1.stl		8.3kB
octa_loop2.stl		33kB
octa_loop3.stl		132kB
octa_loop4.stl		526kB
octa_loop5.stl G		2.1MB
octa_loop6.stl G		8.6MB
octa_loop7.stl G		33MB

Items with G are not on the repository; they are on Google Drive because of their large file size.

Geometric Metrics

• The surface area of a sphere is $A=4\pi r^2$, and when $r=1$, $A\approx 12.566371$.
• The volume of a sphere is $\frac{4}{3}\pi r^3$, and when $r=1$, $V\approx 4.188790$.

Using the Euler Characteristic for a sphere ($v-e+f=2$), we can verify the progression from the base octahedron ($v=6,e=12,f=8$):

The recursive relationships for a closed triangular mesh:

• Faces: $f_{i+1}=4\times f_i$
• Edges: $e_{i+1}=2e_i+3f_i$
• Vertices: $v_{i+1}=v_i+e_i$

Sculpt Baseline

We created Sculpt baseline meshes with the sculpt_stl_to_inp.py script as

(atmeshenv) ~/autotwin/mesh/src/atmesh> python sculpt_stl_to_inp.py

and create standard views in Cubit with

Cubit>
graphics perspective off  # orthogonal, not perspective view
up 0 0 1  # z-axis points up
view iso # isometric x, y, z camera
quality volume 1 scaled jacobian global draw histogram draw mesh list

to produce the following results:

iter	image	cells	nodes nnp	elements nel	element density nel$/V$
0		35x35x35	8,696	7,343	5,507
1		28x28x28	8,133	6,960	2,365
2		26x26x26	7,833	6,744	1,762
3		26x26x26	7,731	6,672	1,630
4		26x26x26	7,731	6,672	1,600
5		26x26x26	7,731	6,672	1,596
6		26x26x26	7,731	6,672	1,595
7		26x26x26	7,731	6,672	1,595

References

• Octa-Loop Subdivision Scheme (GitHub)
  • Documentation detailing the Octa-Loop scheme, a variant of Loop subdivision optimized for octahedrally refined meshes.
• Subdivision Surfaces Lecture Notes (Stanford University)
  • A comprehensive academic overview of subdivision concepts, including the mathematical foundations of the Catmull-Clark and Loop schemes.
  • Stanford cs468-10-fall Subdivision http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/10_Subdivision.pdf and Google Drive repo copy
• Catmull–Clark Subdivision Surface (Rosetta Code)
  • A technical resource providing algorithmic steps and multi-language code implementations for the Catmull-Clark subdivision process.
• Recursively Generated B-Spline Surfaces (Original Paper)
  • The seminal 1978 paper by Edwin Catmull and James Clark that introduced the method for generating smooth surfaces from arbitrary topological meshes (referenced within the other materials).
• Catmull, E., & Clark, J. (1978). Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design, 10(6), 350-355. https://doi.org/10.1016/0010-4485(78)90110-0
• Loop, C. (1987). Smooth subdivision surfaces based on triangles [Master's thesis, University of Utah]. https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/thesis-1.pdf
• https://docs.juliahub.com/Meshes/FuRcu/0.17.1/algorithms/refinement.html#Catmull-Clark

1. Boyé S, Guennebaud G, Schlick C. Least squares subdivision surfaces. In Computer Graphics Forum 2010 Sep (Vol. 29, No. 7, pp. 2021-2028). Oxford, UK: Blackwell Publishing Ltd. ↩