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{\bf r} = {\bf r}(l)

where

...

= \{ x(l), \, y(l), \, z(l) \}

where

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body{\bf r}(l)

position vector of trajectory points

space coordinates

with 

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bodyz

-ccordinate facing down to the Earth Centreintake pipeline coordinates 

(l)
being a vertical direction

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body

{\bf r}_0

l

path length from pipeline intake point 

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body

\

{

x_s = 0, \, y_s = 0, \, z_s = 0 \}pipeline length from inflow point 

\bf r}_0

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body

l = \int_0^l \sqrt{dx^2 + dy^2 + dz^2} = \int_0^l \sqrt{\dot x^2 + \dot y^2 + \dot z^2} dl

{\bf r}_0

pipeline intake point coordinates 

LaTeX Math Inline
body\{ x_

s

0 = 0, \, y_

s

0 = 0, \, z_

s

0 = 0 \}