Biflippers and head to tail composition rules

TL;DR

The paper introduces a novel graphical calculus using biflippers for visualizing and composing isometries in low-dimensional classical geometries, generalizing vector addition and translation composition.

Contribution

It presents a new biflipper-based graphical calculus that extends traditional vector and translation operations to isometries in low-dimensional spaces.

Findings

Biflippers provide an intuitive visual tool for isometry composition.

The calculus generalizes vector addition and translation composition.

It simplifies understanding of geometric transformations.

Abstract

A new graphical calculus for operating with isometries of low dimensional spaces of classical geometries is proposed. It is similar to a well-known graphical representation for vectors and translations in an affine space. Instead of arrows, we use biflippers, which are arrows framed at the end points with subspaces. The head to tail addition of vectors and composition of translations is generalized to head to tail composition rules for isometries.