Steering Bands for Visualizing Complex Maps: Flow, Composition, Inverses, Wrapping, and Tilted Readout

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Original abstract

This visual note develops a pedagogical “steering-band” method for following complex mappings between separated input and output planes. A band is not introduced as a new mathematical object; it is a visual guide whose fibers make point correspondence easy to trace. The examples show the segment map ti ↦ t = −iz, the flip ti ↦ (1−t)i = i−z, two-stage composition, inversion by reading the same guides backward, wrapping the segment [0,i] once around the unit circle via w = e²πz, and a tilted-output-plane example in which a fixed vertical-drop/readout rule produces the directional stretch x+iy ↦ 2x+iy at a 60° tilt. The central pedagogical idea is that the eye can follow a point through a map, through a composition, and backward through an inverse without relying on unexplained correspondence arrows. The exact interior shape of a steering band is supplementary unless explicitly specified; what matters is which input point is connected to which output point. The note also distinguishes invertible examples from the wrapping example, where two distinct input endpoints reach the same output point. Authorship / AI assistance note The steering-band visualization and its pedagogical use originated with Tomasz Owsiak. ChatGPT assisted in rapidly rendering, numerically checking, and refining the diagrams and formulas.

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