# Geometry of 3D Environments and Sum of Squares Polynomials

arXiv: https://arxiv.org/abs/1611.07369 · HTML: https://mlsanity.com/paper/1611.07369/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Nov 2016.

## Built on by (most cited first, 2 of 2)

- [Learning Polynomial Representations of Physical Objects with Application to Certifying Correct Packing Configurations](https://mlsanity.com/paper/2312.06791/) (2023)
- [A practical, fast method for solving sum-of-squares problems for very large polynomials](https://mlsanity.com/paper/2410.19844/) (2024)

## Abstract (arXiv)

Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an obstacle) with convex or nearly-convex basic semialgebraic sets, computation of Euclidean distances between two such sets, separation of two convex basic semalgebraic sets that overlap, and tight containment of the union of several basic semialgebraic sets with a single convex one. We use algebraic techniques from sum of squares optimization that reduce all these tasks to semidefinite programs of small size and present numerical experiments in realistic scenarios.