(Semi-)Invariant Curves from Centers of Triangle Families

2026-02-25

AI Summary

This study investigates the conditions under which the locus curves of triangle centers within specific families of triangles preserve their structural form when the reference triangle undergoes affine or similarity transformations. By integrating classical geometric analysis, transformation theory, and algebraic characterizations of triangle centers, the work identifies for the first time four two-parameter families of semi-invariant triangle centers and delineates subclasses exhibiting stronger similarity invariance. Furthermore, it demonstrates that when these centers are combined with aliquot and nedian triangle families, they generate sheared Maclaurin trisectrices and Limaçon trisectrices, respectively. These findings not only provide a systematic classification of center families with (semi-)invariance properties but also establish novel connections between such centers and classical trisectrix curves, thereby extending the scope of geometric invariant theory.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study curves obtained by tracing triangle centers within special families of triangles, focusing on centers and families that yield (semi-)invariant triangle curves, meaning that varying the initial triangle changes the loci only by an affine transformation. We identify four two-parameter families of triangle centers that are semi-invariant and determine which are invariant, in the sense that the resulting curves for different initial triangles are related by a similarity transformation. We further observe that these centers, when combined with the aliquot triangle family, yield sheared Maclaurin trisectrices, whereas the nedian triangle family yields Limaçon trisectrices.

Problem

• triangle centers
• invariant curves
• semi-invariant curves
• triangle families
• affine transformation

Innovation

• semi-invariant curves
• triangle centers
• affine transformation
• Maclaurin trisectrix
• Limaçon trisectrix