An Intrinsic Log–Polar Hull Measure for Joint Magnitude–Direction Dispersion in Real-World Streams

Circular variables (e.g., headings, bearings) are inherently branch-cut objects: the same direction may be reported as 359◦ or −1◦. When direction is coupled with a positive magnitude (e.g., wind speed, epicentral distance, tree diameter), standard directional summaries (mean resultant length, circular variance) discard magnitude, while Euclidean summaries ignore the 2π periodicity. This manuscript introduces a new, branch-free dispersion functional—the Sakib Index (SI)—defined from the convex hull area of points in log–polar coordinates and made intrinsic by optimizing over all possible angle cuts. The core novelty is the S M Nazmuz Sakib Branch-Free Cut Lemma, which shows that the objective is constant on intervals between observed angles, reducing the intrinsic minimization to a finite family of “wrap” configurations. We demonstrate dataset-based computations on (i) NOAA/NDBC real-time buoy winds, (ii) the USGS earthquake CSV feed in a regional window, and (iii) a Socrata/NYC Open Data sample of street-tree diameters (DBH) augmented only for visualization density. Empirically, SI behaves as an interpretable “joint exposure” score: lower SI corresponds to larger branch-free log–polar hull area, indicating simultaneous spread in magnitude and direction.