Ill-Posed Configurations in Random and Experimental Data Points Collection

TL;DR

This paper investigates the occurrence of ill-posed geometric configurations in both random and physical data, revealing higher degeneracy risks in real-world systems due to systematic biases, which impacts computational geometry tasks.

Contribution

It introduces a probabilistic framework analyzing how data collection methods influence the likelihood of degeneracies in geometric configurations.

Findings

Degeneracies are more frequent in physical systems than in random data.

Systematic biases in data collection increase the risk of ill-posed configurations.

Understanding these probabilities aids in designing more robust geometric algorithms.

Abstract

Ill-posed configurations, such as collinear or coplanar point arrangements, are a persistent challenge in computational geometry, complicating tasks as in triangulation and convex hull construction. This paper discusses the probability of such configurations arising in two scenarios: (1) data sampled randomly from a uniform distribution, and (2) data collected from physical systems, such as reflective surfaces or structured environments. We present a probabilistic framework, analyze the geometric and sampling constraints, and provide some mathematical insights into how data acquisition processes influence the likelihood of degeneracies. Notably, our findings reveal that degeneracies occur more frequently in physical systems than in purely random simulations due to systematic biases introduced by instrumental setups and environmental structures, emphasizing the risks of drawing conclusions solely based on assumptions derived from random data.