Affine congruence by dissection of discs - appropriate groups and optimal dissections

Abstract

Let \( \mathcal{G} \) be a group of affine transformations of the Euclidean plane \( {\mathbb{R}}^2 \). Two topological discs D, \( {\rm E} \subseteq \mathbb{R}^{2} \) are called congruent by dissection with respect to \( \mathcal{G} \)if D can be dissected into a finite number of subdiscs that can be rearranged by maps from \( \mathcal{G} \) to a dissection of E.