Construction of complete (k,3)-Arcs containing menelaus 6-Figures in PG(2,4) using a secant-based algorithm

Year 2025, Issue: 014, 16 - 23, 30.12.2025

Ayşe Bayar , Elif Altıntaş Kahriman

Abstract

This study investigates complete (k,3)-arcs containing a Menelaus 6-figure in the projective plane PG(2,4). We developed and implemented a computational algorithm in C# to construct and classify complete arcs through a detailed analysis of the secant line distributions associated with the Menelaus 6-figure. The algorithm identifies seven points outside the Menelaus 6-figure, which are classified into two categories based on their incidence with 0-secant, 1-secant, and 2-secant lines. By analyzing the extension of this configuration, one unique complete (7,3)-arc and eight distinct complete (9,3)-arcs were identified, all of which contain the Menelaus 6-figure.

Keywords

projective plane , (k,3)-arc , Menelaus 6-figure , finite geometry , PG(2,4)

PG(2,4) Düzleminde Menelaus 6-figür İçeren Tam (k,3)-Arkların Sekant-Tabanlı Bir Algoritma ile Oluşturulması

Abstract

Bu çalışma, projektif düzlem PG(2,4)’te Menelaus 6-figürünü içeren tam (k,3)-arkları incelemektedir. C# dilinde geliştirilen ve uygulanan hesaplamalı bir algoritma aracılığıyla, Menelaus 6-figüre ait sekant doğru dağılımlarının ayrıntılı analizi yapılarak tam arkların inşası ve sınıflandırılması gerçekleştirilmiştir. Algoritma, Menelaus 6-figürün dışında bulunan yedi noktayı belirlemekte ve bu noktaları 0-sekant, 1-sekant ve 2-sekant doğrularıyla ilişkilerine göre iki ayrı kategoriye ayırmaktadır. Bu konfigürasyonun genişlemesi incelendiğinde, Menelaus 6-figürü içeren bir adet özgün tam (7,3)-ark ve sekiz farklı tam (9,3)-ark tespit edilmiştir.

Keywords

projektif düzlem , (k,3)-ark , Menelaus 6-figür , sonlu geometri , PG(2,4)

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