The Digital Involute: A Discrete Alignment Method for the Inverse Involute Function
陈典章1
作者信息
1. 独立研究者
折叠
摘要
传统反渐开线函数求解依赖迭代或级数展开，面临初值敏感、分段缝合、多精度仿真无法在有限字长硬件上物理复现等困境。本文给出数字渐开线：以曲率域渐开线方程 1/tana=1/(k+a)为数学本体，以归正方程 U(1+UK)=D为工程形式，在 IEEE 754 硬件上完成归正的投影过程——离散，终态即硬件最小不可分辨的离散坐标。
给出固定顺序的初等运算链（双精度工程链与四精度基准链），用于在给定展角 k=tana-a下直接生成渐开线齿廓的极坐标 (θ,ρ)。该链不含分支、迭代、截断或人为阈值，无外部参考，全域适用，时延固定（一次 arctan、一次 tan），精度仅由浮点数舍入误差决定。前向代入检验 1/tana ⊖ 1/(k+a)=0由硬件零标志位锁定终态唯一。
双精度下误差谱表明：归真输出 a 始终锁定在硬件邻域内，在 k>10^(-4)时参考误差进入 10^(-15)rad 量级，深零域邻域半径峰值约 2.1×10^(-13)rad；增强速算 W 在 k<0.36（α<52^∘）精度达硬件极限，相对于高精度参考真值的全域最大误差约 6×10^(-12)rad；轻量速算 M 在 k<15（α<86^∘）优于 5×10^(-9)rad，覆盖工控需求。
数字渐开线是理论渐开线在浮点网格上的离散投影点列，可作为齿轮设计、制造、计量的数字基准路径，不依赖物理展成链。
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. Keywords: digital involute, curvature-driven, alignment equation, hardware projection, discretization, digital reference, CNC
关键词
数字渐开线，曲率驱动，归正方程，硬件投影，离散，数字基准，数控
Key words
digital involute/ curvature-driven/ alignment equation/ hardware projection/ discretization/ digital reference/ CNC
