Computational Validation of the Oloid as a Local Optimum in the Developable Roller Family

TL;DR

This paper introduces the Contact Distribution Score (CDS) and Stress Distribution Score (SDS) to evaluate how uniformly convex bodies distribute contact during rolling, identifying the oloid as a near-optimal shape with significant advantages.

Contribution

It develops a formal metric system and computational framework to analyze geometric forms as engineering substrates, highlighting the oloid's optimal contact distribution.

Findings

The oloid significantly outperforms the cylinder baseline in contact and stress distribution scores.

The oloid's geometry contributes minimally to stress non-uniformity beyond contact distribution.

The oloid maintains a consistent advantage across fatigue, thermal, and wear scores.

Abstract

Many engineering failures (thermal hotspot concentration, Hertz contact fatigue localization, boundary-layer loss, mixing dead zones) are geometric failure modes: changing the material delays the failure; changing the geometry eliminates it. Despite this, no formal metric exists for evaluating how uniformly a convex body distributes surface contact during rolling, with direct engineering implications. We introduce the Contact Distribution Score (CDS), a scalar metric defined as the area-weighted variance of contact time over a rolling surface, and its stress-domain counterpart the Stress Distribution Score (SDS), the area-weighted variance of accumulated Hertz contact pressure. CDS -> 0 indicates uniform contact; SDS -> 0 indicates uniform stress. We implement a three-layer oracle architecture (approximate oracle for search, rigid-body oracle for validation, Hertz contact pressure oracle for SDS). A parametric search over 45 members of the developable roller family identifies the oloid (Schatz, 1929) at CDS = 8.2 x 10^-7, with the conventional cylinder baseline at 4.75 x 10^-5: a 58x discrimination. Independent curvature-driven analysis under uniform contact yields a geometry-only SDS of 4.8 x 10^-8, indicating the oloid's surface curvature contributes minimal additional stress non-uniformity beyond the contact distribution. We extend the analysis to fatigue (FDS), thermal (TDS), and wear (WDS) scores, finding the oloid's 58x advantage transfers consistently across linear and multiplicative metrics in a 46-68x range. The nonlinear fatigue metric diverges due to Basquin S-N amplification but still shows oloid superiority over all tested alternatives. This work establishes the formal vocabulary and computational infrastructure for substrate geometry: the study of geometric forms as engineering substrates classified by their operational invariants.