Minimum-Area Enclosing Triangle with a Fixed Angle

TL;DR

This paper presents an efficient algorithm for finding the minimum-area triangle with a fixed angle that encloses a set of points, with specialized fast solutions for convex polygons and complexity bounds.

Contribution

It introduces an O(n log n) algorithm for the problem, proves the necessity of cubic roots in general solutions, and establishes a lower bound, with a linear-time algorithm for convex polygons.

Findings

O(n log n) algorithm for general point sets

Linear-time algorithm for convex polygons

Proven lower bound of Omega(n log n) in the algebraic model

Abstract

Given a set S of n points in the plane and a fixed angle 0 < omega < pi, we show how to find in O(n log n) time all triangles of minimum area with one angle omega that enclose S. We prove that in general, the solution cannot be written without cubic roots. We also prove an Omega(n log n) lower bound for this problem in the algebraic computation tree model. If the input is a convex n-gon, our algorithm takes Theta(n) time.