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arXiv · 2102.04633

$k$-Equivalence Relations and Associated Algorithms

Daniel Selsam·Jesse Michael Han
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Abstract

Lines and circles pose significant scalability challenges in synthetic geometry. A line with $n$ points implies ${n \choose 3}$ collinearity atoms, or alternatively, when lines are represented as functions, equality among ${n \choose 2}$ different lines. Similarly, a circle with $n$ points implies ${n \choose 4}$ cocyclicity atoms or equality among ${n \choose 3}$ circumcircles. We introduce a new mathematical concept of $k$-equivalence relations, which generalizes equality ($k=1$) and includes both lines ($k=2$) and circles ($k=3$), and present an efficient proof-producing procedure to compute the closure of a $k$-equivalence relation.

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Daniel Selsam, Jesse Michael Han. 2021-02-09. $k$-Equivalence Relations and Associated Algorithms. https://arxiv.org/abs/2102.04633

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