We investigate the origin of a unified scaling relation in spiral galaxies .
Observed spiral galaxies are spread on a plane in the three-dimensional logarithmic space of luminosity L , radius R and rotation velocity V .
The plane is expressed as L \propto ( VR ) ^ { \alpha } in I -passband , where \alpha is a constant .
On the plane , observed galaxies are distributed in an elongated region which looks like the shape of a surfboard .
The well-known scaling relations , L - V ( Tully-Fisher relation ) , V - R ( also the Tully-Fisher relation ) and R - L ( Freeman ’ s law ) , can be understood as oblique projections of the surfboard-like plane into 2-D spaces .
This unified interpretation of the known scaling relations should be a clue to understand the physical origin of all the relations consistently .
Furthermore , this interpretation can also explain why previous studies could not find any correlation between TF residuals and radius .

In order to clarify the origin of this plane , we simulate formation and evolution of spiral galaxies with the N -body/SPH method , including cooling , star formation and stellar feedback .
Initial conditions are set to isolated 14 spheres with two free parameters , such as mass and angular momentum .
The CDM ( h = 0.5 , \Omega _ { 0 } = 1 ) cosmology is considered as a test case .
The simulations provide the following two conclusions : ( a ) The slope of the plane is well reproduced but the zero-point is not .
This zero-point discrepancy could be solved in a low density ( \Omega _ { 0 } < 1 ) and high expansion ( h > 0.5 ) cosmology .
( b ) The surfboard-shaped plane can be explained by the control of galactic mass and angular momentum .