(Semi-)Invariant Curves from Centers of Triangle Families
2026-02-25 • Computational Geometry
Computational Geometry
AI summaryⓘ
The authors study special points inside triangles and the shapes made when these points move as the triangle changes. They focus on points and triangle families that produce shapes which only stretch or rotate the same way across different triangles. The researchers find specific groups of these points that behave this way and identify which are fully consistent under size changes. They also discover that combining these points with certain triangle setups makes shapes called trisectrices with known names.
triangle centerslociaffine transformationsimilarity transformationMaclaurin trisectrixLimaçon trisectrixtriangle familiesaliquot trianglenedian triangle
Authors
Klara Mundilova, Oliver Gross
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.