v

along

x

, the velocity-addition formula formulas are

u_{x}^{'} = \frac{u _{x} - v}{1 - u _{x} v / c ^{2}}, u_{y}^{'} = \frac{u _{y}}{γ _{v} ( 1 - u _{x} v / c ^{2} )} .

(1)

Substitute

u_{x} = u cos θ

u_{y} = u sin θ

and take the ratio. The common time-transformation denominator cancels, yielding

tan θ^{'} = \frac{u sin θ}{γ _{v} ( u cos θ - v )} .

(2)

To specify the direction rather than just its tangent, use

θ^{'} = atan2 (u sin θ / γ_{v}, u cos θ - v)

. This keeps the correct quadrant when the boost reverses the sign of the longitudinal velocity. For physical subluminal speeds the canceled denominator is positive. At

u_{x} = v

and nonzero

u_{y}

, the transformed direction is perpendicular to the boost; if the particle is exactly comoving, its transformed velocity vanishes and the angle is undefined.