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*-axis measures time,

_{1}*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*-axis is dependent, it measures length,

_{1}*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*-axis is dependent, it measures time,

_{1}*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}.