Distances between realizations of order types

TL;DR

This paper investigates the complexity of transforming point configurations in the plane by continuous motions, establishing bounds on the number of order type changes needed, which are tight for configurations of the same order type.

Contribution

It proves that any two point configurations can be connected through at most inom{n}{3} order type changes, and this bound is sharp even for configurations sharing the same order type.

Findings

Maximum of inom{n}{3} intermediate changes needed for transformations.

Existence of pairs requiring the maximum number of changes.

Sharpness of the cubic bound for same order type configurations.

Abstract

Any $n$-tuple of points in the plane can be moved to any other $n$-tuple by a continuous motion with at most $\binom{n}{3}$ intermediate changes of the order type. Even for tuples with the same order type, the cubic bound is sharp: there exist pairs of $n$-tuples of the same order type requiring $c\binom{n}{3}$ intermediate changes.