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