This continues the previous post here.
The parametric equation for a circular helix around the x1-axis with radius r and slope b/a (or pitch 2πb) is x1(t) = bt, x2(t) = r cos(t), x3(t) = r sin(t). Its arc length equals t · sqrt(r² + b²).
The parametric equation for a circular helix around the t1-axis with radius q and slope c/q (or pitch 2πc) is t1(x) = cx, t2(x) = a cos(x), t3(x) = q sin(x). Its arc length equals x · sqrt(q² + c²).
The linear motion along the x1-axis measures length, s, with velocity b. The circular motion around the x1-axis measures time, t, as an angle. The linear motion along the t1-axis measures length, w, with velocity c. The circular motion around the t1-axis measures length, x, as an angle.
(1) If the circular motion around the x1-axis is independent, it measures time, t, as an angle. If the linear motion along the x1-axis is dependent, it measures length, s, as a parameter, and the axis is a space axis, xs.
(2) If the linear motion along the x1-axis is independent, it measures length, x, as a parameter, with velocity b. If the circular motion around the x1-axis is dependent, it measures time, t, as an angle, and the axis is a time axis, xt.
(3) If the motion is helical, that is both circular and linear, then if it is measured by length, it forms a curve in 3D space. If it is measured by time, it forms a curve in 3D time.
Galilean transformation with independent time: xt´ = xt, xs´ = xs – vx. xt, ys´ = ys, , zs´ = zs.
Simplified notation: t´ = t, x´ = x – vt, y´ = y, z´ = z.
Galilean transformation with independent space: xs´ = xs, xt´ = xt – ux. xs, yt´ = yt, , zt´ = zt.
Simplified notation: s´ = s, t´ = t – ux, t2´ = t2, t3´ = t3.