clers

CBEAD/CLERS encode/decode and canonical naming for triangulations of the sphere.

This repository contains small standalone C and Python implementations of a CLERS-style traversal code for sphere triangulations. In the code and examples here, the alphabet is written in the CBEAD variant rather than the classical CLERS lettering used in the Edgebreaker literature.

The correspondence is:

• classical C L E R S
• here C B E A D

So references to CLERS in the literature correspond here to CBEAD in the code.

For us, this traversal code is useful not just as a compact encoding, but as an assembly order for actually building a polyhedron face by face.

The implementation here is in the Edgebreaker / corner-table tradition, but the emphasis is different from production mesh codecs: exact encode/decode for triangulations of the sphere, canonical names, and simple batch tools.

What is here

• src/clers.c:
  • batch command-line tool
  • decodes CBEAD/CLERS names to face lists
  • encodes face lists to traversal names
  • computes canonical traversal names
• src/clers.py:
  • reference-quality Python implementation
  • iterative encode/decode
  • canonical naming in both oriented and unoriented forms

Alphabet

This repository uses the letters

• C
• B
• E
• A
• D

which correspond to the classical Edgebreaker / CLERS letters

• C
• L
• E
• R
• S

respectively.

Scope

Input objects are triangulations of the sphere. A triangulation is represented as a list of triples (a,b,c) with vertices labeled 1..n, and each oriented edge (a,b) belongs to exactly one face (a,b,c).

C tool

Compile with:

make

This builds bin/clers. The executable supports four modes:

bin/clers decode
bin/clers encode
bin/clers name
bin/clers canonical

Meanings:

• decode: traversal name -> face list
• encode: face list -> one traversal name
• name: face list -> canonical traversal name
• canonical: traversal name -> canonical traversal name

Input is one object per line on stdin.

Face-list format is:

a,b,c;d,e,f;...

Example:

echo '1,2,3;1,3,4;1,4,2;2,4,3' | bin/clers name

Python module

The Python file at src/clers.py provides the main functions:

• encode(poly, start=None)
• decode(recipe, verify=False)
• official_name(poly)
• official_unoriented_name(poly)

The implementations are iterative rather than recursive.

Example (from the repo root, with src/ on PYTHONPATH):

import sys; sys.path.insert(0, "src")
from clers import encode, decode, official_name, official_unoriented_name

tet = [(1,2,3),(1,3,4),(1,4,2),(2,4,3)]

name = official_unoriented_name(tet)
poly = decode(name, verify=True)

print(name)
print(poly)

Canonical naming

Canonical names are obtained by minimizing over starts at minimum-degree vertices; in the unoriented case, both orientations are considered.

This makes the code useful not only for encode/decode, but also for:

• naming triangulated spheres
• deduplicating generated nets
• detecting isomorphism of embedded triangulations
• maintaining stable names in enumeration pipelines

Why this repo exists

A spiral code is attractive when it works, but it breaks down as a general naming scheme. CBEAD/CLERS gives a compact local traversal code that is easy to decode, useful for actual face-by-face construction, and flexible enough to support canonical naming.

That combination makes it a good fit for triangulated-sphere workflows in which one wants all of the following:

• generation
• encode/decode
• canonical names
• isomorphism detection
• human-readable build order

Notes

This repository is intentionally small and transparent. It is not trying to be a full industrial mesh-compression library. The point is to have a simple, exact, hackable implementation for triangulations of the sphere.

Keywords

CBEAD, CLERS, Edgebreaker, corner table, triangulation, sphere triangulation, canonical naming, canonical labeling, graph encoding, polyhedra.

Provenance

The code and documentation in this repository were drafted primarily with Claude Code under the author's direction, with additional advice, review, and supervision from ChatGPT.