From: g.scholten@gmx.de   
      
   Hi all,   
      
   I recently asked myself by which factor a rod is lorentz-contracted that   
   is neither oriented in the direction of movement (typically x-direction)   
   nor in a perpendicular direction, but in a direction in between, e.g.   
   rotated by some angle against the x-direction in the (x,y) plane.   
      
   Since I did not find any resource about this that quickly, I examined   
   the result on my own:   
      
   Let phi be the angle between the rod's orientation direction and the   
   x-axis in the rod's rest frame S, then we have in S   
      
   Lx = L cos(phi)   
   Ly = L sin(phi)   
   L = (Lx^2 + Ly^2)^(1/2)   
      
   where L is the total length of the rod in S and Lx und Ly the particular   
   length in x- and y-direction.   
      
   Now consider a frame S' in which the rod is moving in x'-direction with   
   velocity v'. Then there are two questions coming up:   
      
   - What is the rod's length L' in S'?   
   - What is the orientation angle phi' against x'-axis in S'?   
      
   Obviously, the particular lengths Lx' und Ly' in S' in x'- and   
   y'-direction obey   
      
   Lx' = Lx / gamma = (1 - (v'/c)^2)^(1/2) Lx   
   Ly' = Ly   
      
   In other words: in x'-direction, the rod is simply lorentz-contracted by   
   factor 1/gamma, in y-direction, the rod's length is unmodified. This   
   yields for the total length L' in S'   
      
   L' = (Lx'^2 + Ly'^2)^(1/2)   
       = [(Lx^2 / gamma^2) + Ly^2]^(1/2)   
       = L [(cos(phi)^2 / gamma^2) + sin(phi)^2]^(1/2)   (1)   
      
   We can check this result by comparing to the well-known results for phi   
   = 0 (rod oriented in x-direction) and phi = pi/2 (rod oriented in   
   y-direction):   
      
   phi = 0: L' = L (1 / gamma^2)^(1/2) = L / gamma  -> OK   
   phi = pi/2: L' = L   -> OK   
      
   Now what is the angle phi' in S'? We get   
      
   cos(phi') = Lx' / L'   
              = (Lx / gamma) / [L ((cos(phi)^2 / gamma^2)   
                                    + sin(phi)^2)^(1/2)]   
              = (Lx / gamma) / (Lx / gamma^2 + Ly^2)^(1/2)   
              = Lx / (Lx + gamma^2 Ly^2)^(1/2)                     (2a)   
      
   Or expressed by phi:   
      
   cos(phi') = cos(phi) / (cos(phi)^2 + gamma^2 sin(phi)^2)^(1/2)  (2b)   
      
   We again check this:   
      
   phi = 0:   
       cos(phi') = cos(0) / (cos(0)^2 + gamma^2 sin(0)^2)^(1/2)   
                 = 1  => phi' = 0          -> OK   
   phi = pi/2:   
       cos(phi') = cos(pi/2) / (cos(pi/2)^2 + gamma^2 sin(pi)^2)^(1/2)   
                 = 0  => phi' = pi/2       ->   
      
   For phi < pi/2, i.e. cos(phi) > 0, we can re-write (2b) to   
      
   cos(phi') = 1 / (1 + gamma^2 tan(phi)^2)^(1/2)   (2c)   
      
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