Why the optic axis of an crystal should has a proper direction; why it cannot be a single line? In many books it is written that the optic axis of a crystal is a direction rather than a single line! I really cannot understand what is the necessity of the direction and why I cannot say that it has the opposite direction rather than what is defined in the book?
 A: The optic axis is just a line. It does have a direction but the opposite direction has still the same direction.
The meaning of "direction" is somewhat unclear. The wind can blow out of the eastern direction. This means though that it's not coming precisely from the east though. But in this case, it's precisely defined though. why I cannot say that it has the opposite direction rather than what is defined in the book? Actually, you can. I guess for the math this introduces a minus sign. The line can be placed anywhere in the crystal (if it's a homogeneous, isomorphic crystal).
A: A birefringent (optically active) crystal, like iceland spar (calcite),
does not host a light ray with the same wave-propogation character
as an isotropic glass.
To trace a light ray which impinges on a surface, one needs to know
not only the angle to the surface, but the angle to the optic axis
(which is a coordinate axis of the crystal geometry).
Thus, the 'axis' reference is to a second vector direction in space,
in addition to the direction of the normal vector of the surface, and
the propogation direction of the incoming light.   It
is not at all like an axle, which has a location as well as an
orientation direction.
