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$P'_x = R\ \bigl( cos( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) - sin( P_a )\ sin( \frac{ v\ \Delta t }{ R } \bigr)\ \bigr) \\ P'_y = R\ cos( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ P'_z = R\ sin( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr)$

$P_x = R,\ P_y = P_z = 0,\ v_x = v_z = 0$

$S$

$cos( S_a ) = 1 \\ sin( S_a ) = 0 \\ cos( P_a ) = 1 \\ sin( P_a ) = 0 \\$

$\begin{align} P'_x &= R\ \bigl( cos( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) - sin( P_a )\ sin( \frac{ v\ \Delta t }{ R } \bigr)\ \bigr) \\ &= R\ cos( \frac{ v\ \Delta t }{ R } \bigr) \\ P'_y &= R\ cos( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ &= R\ sin( \frac{ v\ \Delta t }{ R } \bigr) \\ P'_z &= R\ sin( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ &= 0 \end{align}$

$P_y = 0,\ v_y = 0$

$S$

$cos( S_a ) = 0 \\ sin( S_a ) = 1 \\ cos( P_a ) = \frac{ P_x }{ R } \\ sin( P_a ) = \frac{ P_z }{ R } \\$

$\begin{align} P'_x &= R\ \bigl( cos( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) - sin( P_a )\ sin( \frac{ v\ \Delta t }{ R } \bigr)\ \bigr) \\ &= R\ \bigl( \frac{ P_x }{ R }\ cos( \frac{ v\ \Delta t }{ R } ) \bigr) - \frac{ P_z }{ R }\ sin( \frac{ v\ \Delta t }{ R } \bigr) \bigr) \\ &= P_x\ cos\bigl( \frac{ v\ \Delta t }{ R } \bigr) - P_z\ sin\bigl( \frac{ v\ \Delta t }{ R } \bigr) \\ \ \\ P'_y &= R\ cos( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ &= 0 \\ \ \\ P'_z &= R\ sin( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ &= R\ \bigl( \frac{ P_z }{ R }\ cos( \frac{ v\ \Delta t }{ R } ) + \frac{ P_x }{ R }\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ &= P_z\ cos\bigl( \frac{ v\ \Delta t }{ R } \bigr) + P_x\ sin\bigl( \frac{ v\ \Delta t }{ R } \bigr) \end{align}$

$P_x = R,\ P_y = P_z = 0$

$P$

$cos( S_a ) = \frac{ v_y }{ v } \\ sin( S_a ) = \frac{ v_z }{ v } \\ cos( P_a ) = 1 \\ sin( P_a ) = 0 \\$

$\begin{align} P'_x & = R\ \bigl( cos( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) - sin( P_a )\ sin( \frac{ v\ \Delta t }{ R } \bigr)\ \bigr) \\ & = R\ \bigl( 1\ cos( \frac{ v\ \Delta t }{ R } ) - 0 \bigr)\ \bigr) \\ &= R\ cos( \frac{ v\ \Delta t }{ R } \bigr) \\ \ \\ P'_y & = R\ cos( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ & = R\ \frac{ v_y }{ v }\ \bigl( 0 + 1 \ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ & = R\ \frac{ v_y }{ v }\ \bigl( sin( \frac{ v\ \Delta t }{ R } ) \bigr)\ \\ \ \\ P'_z & = R\ sin( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ & = R\ \frac{ v_z }{ v }\ \bigl( sin( \frac{ v\ \Delta t }{ R } ) \bigr)\ \\ & = R\ \frac{ v_z }{ v }\ \bigl( sin( \frac{ v\ \Delta t }{ R } ) \bigr)\ \\ \end{align}$

$cos( S_a ) = \frac{ P_y }{ \sqrt{ R^2 - P_x^2 } } \\ sin( S_a ) = \frac{ P_z }{ \sqrt{ R^2 - P_x^2 } } \\ cos( P_a ) = \frac{P_x }{R } \\ sin( P_a ) = \frac{ \sqrt{ R^2 - P_x^2 } }{ R } \\$

$\begin{align} P'_x &= R\ \bigl( cos( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) - sin( P_a )\ sin( \frac{ v\ \Delta t }{ R } \bigr) \ \bigr) \\ &= R\ \bigl( \frac{P_x }{R }\ cos( \frac{ v\ \Delta t }{ R } ) - \frac{ \sqrt{ R^2 - P_x^2 } }{ R }\ sin( \frac{ v\ \Delta t }{ R } \bigr) \ \bigr) \\ &= P_x\ cos( \frac{ v\ \Delta t }{ R } ) - \sqrt{ R^2 - P_x^2 }\ sin( \frac{ v\ \Delta t }{ R } \bigr) \\ \ \\ P'_y &= R\ cos( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ &= R\ \frac{ P_y }{ \sqrt{ R^2 - P_x^2 } }\ \bigl( \frac{ \sqrt{ R^2 - P_x^2 } }{ R }\ cos( \frac{ v\ \Delta t }{ R } ) + \frac{P_x }{R }\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ &= P_y\ cos( \frac{ v\ \Delta t }{ R } ) + \frac{ P_x\ P_y }{ \sqrt{ R^2 - P_x^2 } }\ sin( \frac{ v\ \Delta t }{ R } ) \\ \ \\ P'_z &= R\ sin( S_a )\ \bigl( sin( P_a )\ cos( \frac{ v\ \Delta t }{ R } ) + cos( P_a )\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ &= R\ \frac{ P_z }{ \sqrt{ R^2 - P_x^2 } }\ \bigl( \frac{ \sqrt{ R^2 - P_x^2 } }{ R }\ cos( \frac{ v\ \Delta t }{ R } ) + \frac{P_x }{R }\ sin( \frac{ v\ \Delta t }{ R } ) \bigr) \\ &= P_z\ cos( \frac{ v\ \Delta t }{ R } ) + \frac{ P_z\ P_x }{ \sqrt{ R^2 - P_x^2 } }\ sin( \frac{ v\ \Delta t }{ R } ) \\ \end{align}$