Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 13 min ago
PreprintCC BY 4.0arXiv

When Pedal Shapes Come Back: a Complex Story

  • Michał Zwierzyński
Open at sourcePDFDOICite

Abstract

Under the usual nondegeneracy assumptions, repeatedly projecting a fixed point onto the side lines of an n-gon returns its shape after n steps. We explain this classical theorem through a short complex-number identity: the multipliers of one step move cyclically around the polygon. The same calculation allows a different common angle of projection at each step. A sharp angle-dependent contraction bound identifies the incircle centre as the maximiser for tangential polygons. In space, even a regular tetrahedron can fail to recover its shape after four steps. We give an exact example, then show how concentric inscribed and circumscribed spheres restore a two-step return. Rectangular boxes provide a further four-step family.

Related

  1. When Geometry Shapes Information in Pursuit-Evasion Games
    Hélène Le Cadre, Ava Sedighi, Frédy Pokou
    HALPreprintCC BY 4.0Full textOptimisation & controlHAL
  2. Periodic Environmental Forcing Shapes the Stability of Complex Ecological Networks
    Sayantan Nag Chowdhury
    arXivPreprintCC BY 4.0Full textDynamical systemsStatistical & nonlinear physicsarXiv
  3. Improved computation of polynomial roots over number fields when using complex embeddings
    Andrea Lesavourey, Thomas Plantard, Willy Susilo
    Journal of Computational AlgebraPeer-reviewedCC BY-NC1 citedAlgebra & number theoryHAL
  4. Pedal Curves of a Bicorn
    Thierry Dana-Picard, Moshe Hanau, Shmuel Krichevsky
    arXivPreprintCC BY-NC-SA 4.0Full textAlgebra & number theoryarXiv
  5. Come for the vibe, stay for the math
    Hugo Parlier, Bruno Teheux
    arXivPreprintarXiv licenceFull textGeneral mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{zwierzynski2026when,
  title = {{When Pedal Shapes Come Back: a Complex Story}},
  author = {Michał Zwierzyński},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01247},
  eprint = {2611.01247},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01247},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - When Pedal Shapes Come Back: a Complex Story
AU  - Michał Zwierzyński
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01247
UR  - https://arxiv.org/abs/2611.01247
AB  - Under the usual nondegeneracy assumptions, repeatedly projecting a fixed point onto the side lines of an n-gon returns its shape after n steps. We explain this classical theorem through a short complex-number identity: the multipliers of one step move cyclically around the polygon. The same calculation allows a different common angle of projection at each step. A sharp angle-dependent contraction bound identifies the incircle centre as the maximiser for tangential polygons. In space, even a regular tetrahedron can fail to recover its shape after four steps. We give an exact example, then show how concentric inscribed and circumscribed spheres restore a two-step return. Rectangular boxes provide a further four-step family.
ER  -