This continues the previous post *here*.

The parametric equation for a circular helix around the *x*_{1}-axis with radius *r* and slope *b*/*a* (or pitch 2π*b*) is *x*_{1}(*t*) = *bt*, *x*_{2}(*t*) = *r* cos(*t*), *x*_{3}(*t*) = *r* sin(*t*). Its arc length equals *t* · sqrt(*r*² + *b*²).

The parametric equation for a circular helix around the *t*_{1}-axis with radius *q* and slope *c*/*q* (or pitch 2π*c*) is *t*_{1}(x) = *cx*, *t*_{2}(*x*) = *a* cos(*x*), *t*_{3}(*x*) = *q* sin(*x*). Its arc length equals *x* · sqrt(*q*² + *c*²).

The linear motion along the *x*_{1}-axis measures length, *s*, with velocity *b*. The circular motion around the *x*_{1}-axis measures time, *t*, as an angle. The linear motion along the *t*_{1}-axis measures length, *w*, with velocity *c*. The circular motion around the *t*_{1}-axis measures length, *x*, as an angle.

(1) If the circular motion around the *x*_{1}-axis is independent, it measures time, *t*, as an angle. If the linear motion along the *x*_{1}-axis is dependent, it measures length, *s*, as a parameter, and the axis is a space axis, *x*_{s}.

(2) If the linear motion along the *x*_{1}-axis is independent, it measures length, *x*, as a parameter, with velocity *b*. If the circular motion around the *x*_{1}-axis is dependent, it measures time, *t*, as an angle, and the axis is a time axis, *x*_{t}.

(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: *x*_{t}´ = *x*_{t}, *x*_{s}´ = *x*_{s} – *v*_{x}. *x*_{t}, *y*_{s}´ = *y*_{s}, , *z*_{s}´ = *z*_{s}.

Simplified notation: *t*´ = *t*, *x*´ = *x* – *vt*, *y*´ = *y*, *z*´ = *z*.

Galilean transformation with independent space: *x*_{s}´ = *x*_{s}, *x*_{t}´ = *x*_{t} – *u*_{x}. *x*_{s}, *y*_{t}´ = *y*_{t}, , *z*_{t}´ = *z*_{t}.

Simplified notation: *s*´ = *s*, *t*´ = *t* – *ux*, *t*_{2}´ = *t*_{2}, *t*_{3}´ = *t*_{3}.