PaperlayerView exact revision on arXiv

Source-linked AI summary

The longest-edge bisection algorithm may produce degenerating tetrahedra

Sergey Korotov

arXiv:2608.23139v1math.NAcs.NAmath.CO

TL;DR

The paper asks whether unrestricted longest-edge bisection can guarantee nondegenerating tetrahedra. It constructs an explicit admissible sequence, which violates shape regularity and both tetrahedral angle conditions; the example does not imply that every fixed tie-breaking convention follows the same orbit.

  • Problem

    Unrestricted longest-edge bisection allows any tied longest edge, but whether this rule guarantees regular tetrahedra is at issue.

    Source: §1 · p. 1, §4 · p. 3

  • Method

    An exact two-bisection recurrence halves a shape parameter while retaining admissibility, and its iteration constructs a tetrahedral sequence.

    Source: §2 · p. 2

  • Results

    The sequence's normalized volume tends to zero, one dihedral angle tends to zero, and another tends to π, violating shape regularity and both angle conditions.

    Source: §3 · p. 3, §5 · p. 4

  • Takeaways & Limitations

    The example shows that unrestricted longest-edge bisection may degenerate, so tie resolution must be included in the algorithm and analyzed in regularity claims.

    Source: §5 · p. 4

    The counterexample does not show that every fixed tie-breaking convention follows the degenerating orbit.

    Source: §4 · p. 3

Abstract

from arXiv · show

An explicit sequence of tetrahedra generated by the longest-edge bisection algorithm is shown to degenerate. The example violates shape regularity and both the minimum- and maximum-angle conditions, demonstrating that arbitrary tie-breaking among longest edges does not guarantee nondegeneration.

1 Introduction

The paper distinguishes unrestricted geometric longest-edge bisection from marked tetrahedral schemes, where marking and tie-breaking are part of the construction. It gives an exact counterexample showing that an admissible tie choice can produce degeneration.

1 Introduction

  • Marked-edge tetrahedral schemes can yield conforming locally refined meshes with finitely many similarity classes, unlike the unrestricted rule considered here.Source: §1 · p. 1
  • In three dimensions, refinement behavior depends on the marking and tie-breaking conventions as well as the geometric selection rule.Source: §1 · p. 1
  • The note gives an exact counterexample for unrestricted tetrahedral bisection, using a recurrent tie; it does not show degeneration under every fixed tie-breaking convention.Source: §1 · p. 1

2 The two-step longest-edge bisection

The section constructs a two-step recurrence for tetrahedra under longest-edge bisection. Starting from E(1), iteration gives a valid sequence, with ties required only at specified steps.

2 The two-step longest-edge bisection

  • The recurrence makes the retained child F(a) congruent to E(a/2), with the relabeled edge sextuple matching that of E(a/2).Source: §2 · p. 2
  • Iterating the recurrence from E(1) generates the sequence, and every arrow is a valid longest-edge bisection.Source: §2 · p. 2
  • Only arrows leaving O(a) require a tie choice; the construction identifies BD and DC as tied for longest in the stated bisection.Source: §2 · p. 2

3 Violation of regularity and angle conditions

The admissible sequence violates shape regularity and both tetrahedral angle conditions. Its normalized-volume ratio and angular gaps decay geometrically, with one dihedral angle tending to zero and another to π.

3 Violation of regularity and angle conditions

  • For T, shape regularity requires |T| ≥ ch^3_T throughout the family, while the minimum- and maximum-angle conditions bound face and interior dihedral angles.The minimum bound is positive, and the maximum bound is smaller than π.Source: §3 · p. 2
  • The admissible sequence violates shape regularity, the minimum-angle condition, and the maximum-angle condition.The normalized volume ratio has no uniform positive lower bound; one interior dihedral angle tends to zero and another tends to π.Source: §3 · p. 3
  • The angle at MC in O(a) is the interior dihedral angle Θ(a), which tends to π as a tends to zero.Source: §3 · p. 3
  • The normalized-volume ratio and both angular gaps decay at the same geometric order 2^-k/2.Source: §3 · p. 3

4 Tie-breaking and conforming partitions

The decisive feature is an exact longest-edge tie, which lets a particular tie choice generate the degenerating sequence. The local chain can also be scheduled within the global face-to-face algorithm, yielding conforming partitions with the same failures.

4 Tie-breaking and conforming partitions

  • The exact equality BD = DC in every O(a) creates the tie that permits the degenerating branch.A deterministic rule selecting the tied edge whose opposite edge is longer selects this branch; this does not mean every fixed tie-breaking convention follows it.Source: §4 · p. 3
  • The rule “choose any longest edge” does not guarantee regularity in three dimensions.Source: §4 · p. 3
  • The local chain can be scheduled inside the global face-to-face longest-edge algorithm, producing conforming partitions with the same three failures.The waiting process is finite because new midpoint-to-opposite-vertex edges are at most (3/2)c when a globally longest edge has length c.Source: §4 · p. 3

5 Conclusion

A closed-form two-bisection recurrence halves a shape parameter while preserving admissibility under the longest-edge rule, yet the resulting infinite sequence degenerates.

5 Conclusion

  • The longest-edge bisection rule may produce degenerating tetrahedra unless tie resolution is included explicitly in the algorithm and analyzed in the regularity theorem.The sequence has normalized volume tending to zero, one dihedral angle tending to zero, and another tending to π.Source: §5 · p. 4
Loading 2608.23139v1…