Ill-Posed Configurations in Random and Experimental Data Points Collection

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.