Torus Starts Here

$\sum { (\Delta a)^{ 2 } } =\sum { (\Delta a_{ l })^{ 2 } } +\sum { (\Delta a_{ t })^{ 2 } }$

Differentiating wrt $t$,

$2\sum { \Delta a } \cfrac { d\Delta a }{ d\, t } =2\sum { \Delta a_{ l } } \cfrac { d\Delta a_{ l } }{ d\, t } +2\sum { \Delta a_{ t } } \cfrac { d\Delta a_{ t } }{ d\, t }$

But,

$\sum { \Delta a } =L_{ or }\quad \sum { \Delta a_{ l } } =L_{ cir }\quad and\quad \sum { \Delta a_{ t } } =n2\pi r_{ p }$

Replacing all the summations,

$L_{ or }\cfrac { d\Delta a }{ d\, t } =L_{ cir }\cfrac { d\Delta a_{ l } }{ d\, t } +n2\pi r_{ p }\cfrac { d\Delta a_{ t } }{ d\, t }$

We know that the velocity components also follows Pythagoras Theorem,

$\left( \cfrac { d\Delta a }{ d\, t }  \right) ^{ 2 }=\left( \cfrac { d\Delta a_{ l } }{ d\, t }  \right) ^{ 2 }+\left( \cfrac { d\Delta a_{ t } }{ d\, t }  \right) ^{ 2 }$

Substitute into the expression,

$L_{ or }\left\{ \left( \cfrac { d\Delta a_{ l } }{ d\, t }  \right) ^{ 2 }+\left( \cfrac { d\Delta a_{ t } }{ d\, t }  \right) ^{ 2 } \right\} ^{ 1/2 }=L_{ cir }\cfrac { d\Delta a_{ l } }{ d\, t } +n2\pi r_{ p }\cfrac { d\Delta a_{ t } }{ d\, t }$

$L_{ or }\left\{ 1+\left( \cfrac { d\Delta a_{ t } }{ d\, t } /\cfrac { d\Delta a_{ l } }{ d\, t }  \right) ^{ 2 } \right\} ^{ 1/2 }=L_{ cir }+n2\pi r_{ p }\left( \cfrac { d\Delta a_{ t } }{ d\, t } /\cfrac { d\Delta a_{ l } }{ d\, t }  \right)$

We can use,

$\cfrac { d\Delta a_{ t } }{ d\, t }=\cfrac{ n2\pi r_{ p } }{T}$

$\cfrac { d\Delta a_{ l } }{ d\, t }=\cfrac{ L_{ cir } }{T}$

which are exact given $T$ is the hypothetical time it takes to travel around the torus making $n$ turns.

$L_{ or }\left\{ 1+\left( \cfrac { d\Delta a_{ t } }{ d\Delta a_{ l } }  \right) ^{ 2 } \right\} ^{ 1/2 }=L_{ cir }+n2\pi r_{ p }\left( \cfrac { d\Delta a_{ t } }{ d\Delta a_{ l } }  \right)$

$L_{ or }\left\{ 1+\left( \cfrac { n2\pi r_{ p } }{ L_{ cir } }  \right) ^{ 2 } \right\} ^{ 1/2 }=L_{ cir }+n2\pi r_{ p }\left( \cfrac { n2\pi r_{ p } }{ L_{ cir } }  \right)$

$L_{ or }^{ 2 }\left\{ 1+\left( \cfrac { n2\pi r_{ p } }{ L_{ cir } }  \right) ^{ 2 } \right\} =L_{ cir }^{ 2 }\left\{ 1+\left( \cfrac { n2\pi r_{ p } }{ L_{ cir } }  \right) ^{ 2 } \right\} ^{ 2 }$

$\left( \cfrac { L_{ or } }{ L_{ cir } }  \right) ^{ 2 }=1+\left( \cfrac { n2\pi r_{ p } }{ L_{ cir } }  \right) ^{ 2 }$

$\left( \cfrac { L_{ or } }{ L_{ cir } }  \right) ^{ 2 }=1+\left( n\cfrac { r_{ p } }{ r_{or}}  \right) ^{ 2 }$

where $L_{ or }$ is the total length of the helix torus, $L_{ cir }$ is the circumference of the orbit, radius $r_{or}$, through the centers of the small circles of the helix, radius $r_p$.  $n$ is the total number of turns of the small circles along the torus.

As if expected,

${ L_{ or } }^{ 2 }=( 2\pi r_{or})^2 +( n2\pi r_{ p } ) ^{ 2 }$

The last relationship is useful for solving torus volume and surface area given $r_p$, $r_{or}$ and $n$.