The outer center-pedal area product in elliptic billiards
- 1Independent researcher
Abstract
For periodic billiards between strictly nested nondegenerate confocal ellipses, the product of the signed areas of the outer tangent polygon and the perpendicular projection of the ellipse center onto its side lines is constant when the period is not divisible by four. The note proves the original k303,b assertion of Reznik-Garcia-Koiller (2020 arXiv;2021 published edition), including admissible coprime stars and fixed repeated traversals, and provides an explicit invariant value. The actual center-foot map is a fixed linear image of a phase-shifted edge-vector polygon. Classical KLS2002 cyclic Jacobi, KS2003/2004 Landen, AST/BT2020 foot-velocity and hodograph, Stachel canonical coordinates, conic polarity and established meromorphic methods are credited. A. Ferudun2026 version1.1, DOI10.5281/zenodo.23088516, Theorem2/Section6, gives the exact complete contemporary theorem. Historical ordering, independent non-derivation and novelty remain undetermined; no first-proof, exclusive-resolution, new-identity or current-worldwide-open claim is made. The deposit includes the concise research note and portable source/verification archive with immutable original submission, scoped audits, exact controls, separately labelled non-interval numerical diagnostics, corrections and reproducibility limitations. Extensive AI tools supported discovery, mathematics, programming, source investigation, writing and internal adversarial checks. Unrefereed preprint; no independent human peer review or formal proof verification is claimed.
Related
- The least-period criterion for outer focal products in elliptic billiards
- A flag-curve proof of an outer-pedal area invariant
- Unbounded orbits in C1-generic centrally symmetric strictly convex outer billiards
- Bounded Orbits in Polygonal Outer Billiards Linked to Divisor Structure — E8 Intelligence Research
- Symmetry-Aware Orientation Estimation via Outer-Product Rotation Targets in YOLO26
BibTeX
@misc{kriebel2026outer,
title = {{The outer center-pedal area product in elliptic billiards}},
author = {Alec Kriebel},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23413614},
url = {https://zenodo.org/records/23413614},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The outer center-pedal area product in elliptic billiards
AU - Alec Kriebel
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23413614
UR - https://zenodo.org/records/23413614
AB - For periodic billiards between strictly nested nondegenerate confocal ellipses, the product of the signed areas of the outer tangent polygon and the perpendicular projection of the ellipse center onto its side lines is constant when the period is not divisible by four. The note proves the original k303,b assertion of Reznik-Garcia-Koiller (2020 arXiv;2021 published edition), including admissible coprime stars and fixed repeated traversals, and provides an explicit invariant value. The actual center-foot map is a fixed linear image of a phase-shifted edge-vector polygon. Classical KLS2002 cyclic Jacobi, KS2003/2004 Landen, AST/BT2020 foot-velocity and hodograph, Stachel canonical coordinates, conic polarity and established meromorphic methods are credited. A. Ferudun2026 version1.1, DOI10.5281/zenodo.23088516, Theorem2/Section6, gives the exact complete contemporary theorem. Historical ordering, independent non-derivation and novelty remain undetermined; no first-proof, exclusive-resolution, new-identity or current-worldwide-open claim is made. The deposit includes the concise research note and portable source/verification archive with immutable original submission, scoped audits, exact controls, separately labelled non-interval numerical diagnostics, corrections and reproducibility limitations. Extensive AI tools supported discovery, mathematics, programming, source investigation, writing and internal adversarial checks. Unrefereed preprint; no independent human peer review or formal proof verification is claimed.
ER -