Largest Empty Rotated Square: lots of trigonometry

2026-05-16 · Source: Yet Another Math Programming Consultant · Field: Science & Research — Mathematics & Computational Sciences, Engineering & Applied Sciences · Depth: Advanced, medium

Summary

This analysis explores methods for finding the largest empty rotated square within a given set of $n$ data points, ensuring the square contains none of the points and remains within an outer bounding box. It extends previous work on axis-aligned and 45-degree rotated squares to arbitrary rotation angles. Two primary cases are examined: when the rotation angle $\theta$ is exogenous (pre-defined) and when it is endogenous (optimized by the model). For the exogenous angle case, the approach involves rotating all data points by $-\theta$ to transform the problem into finding an axis-aligned empty square, then rotating the square's corners back to the original coordinate system to enforce boundary constraints. This results in a linear Mixed-Integer Programming (MIP) model, which for a 25° angle yielded an optimal solution of 3.085938. When the angle is endogenous, the model becomes a Mixed-Integer Non-Linear Programming (MINLP) problem, which Gurobi and Xpress solvers can solve quickly to global optimality, finding a larger square with an area of 9.866 at an angle of 26.728°.

Key takeaway

For Data Scientists working on spatial optimization or geometric pattern recognition, consider the computational benefits of transforming coordinate systems. If your application requires finding the largest empty rotated square, directly optimizing the rotation angle as an endogenous variable within an MINLP model, using solvers like Gurobi or Xpress, will likely yield significantly better results than iterating through fixed angles or interpolating, even for small angular differences.

Key insights

Transforming coordinate systems simplifies finding the largest empty rotated square.

Principles

• Rotate data points, not the square, for exogenous angles.
• Endogenous angle optimization yields superior results.

Method

For exogenous angles, rotate data points by $-\theta$, solve for an axis-aligned square, then rotate the square's corners back to enforce original boundary constraints, forming a linear MIP.

In practice

• Use Gurobi or Xpress for endogenous angle MINLP.
• Avoid linear interpolation for angle optimization.

Topics

• Largest Empty Rotated Square
• Mixed-Integer Programming
• Mixed-Integer Non-Linear Programming
• Geometric Transformation
• Optimization Solvers

Best for: Research Scientist, Data Scientist

Related on AIssential

• Largest empty shapes — Finding largest empty shapes in point sets requires diverse optimization models depending on the shape.
• MINLP instead of indicator constraints? — Replacing indicator constraints with direct multiplication creates competitive MINLP models.
• TriSearch: Learning to Optimize Triangulations via Bistellar Flips — TriSearch uses RL with circuit-supported subtriangulation to optimize polytope triangulations via bistellar flips, generalizing across dimensions.
• Efficient closed-form approaches for pose estimation using Sylvester forms — Sylvester forms enhance resultant-based solvers for faster, accurate pose estimation in computer vision.
• The Discrete-Log Clock: How a Transformer Learns Modular Multiplication — The transformer learns modular multiplication by reducing it to addition in discrete-log space, revealed by using the correct multiplicative character transform.
• A Gentle Introduction to Nonlinear Constrained Optimization with Piecewise Linear Approximations — Linearizing separable nonlinear programs via PWL approximations and SOS type 2 constraints enables efficient solution with LP/MIP solvers.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by Yet Another Math Programming Consultant.