The Digital Involute: A Discrete Alignment Method for the Inverse Involute Function

Abstract

Inverse involute methods rely on iteration or series expansion, with initial-value sensitivity, piecewise stitching, and multi-precision simulation not reproducible on finite-word hardware. We present the digital involute: using curvature-domain equation 1/tan(a)=1/(k+a) as ontology and alignment equation U(1+UK)=D as engineering form, alignment projection onto IEEE 754 hardware (discretization) is completed, terminal state is smallest hardware-resolvable discrete coordinate. A fixed-sequence chain (double-precision engineering and quad-precision reference chains) generates polar coordinates (theta, rho) of involute tooth profile for roll angle k=tan(a)-a. The chain has no branching, iteration, truncation, or threshold, no external reference, global, deterministic latency (one arctan, one tan), precision bounded by floating-point rounding. Forward-substitution test 1/tan(a)-1/(k+a)=0 locks terminal state via hardware zero flag. Under double precision: true-alignment a locked within hardware neighborhood, reference error enters 1E-15 rad for k>1E-4, deep-zero neighborhood radius peak approx 2.1E-13 rad; enhanced W reaches hardware limit for k<0.36 (alpha<52 deg), global max error vs high-precision truth approx 6E-12 rad; lightweight M better than 5E-9 rad for k<15 (alpha<86 deg), covering industrial control. Digital involute is discrete projection point sequence of theoretical involute onto floating-point grid, serving as digital reference path for gear design, manufacturing, metrology, independent of physical generating chains.