What does "Bidirectional" in BRDF mean? So Bidirectional essentially means "two directions", right? But what are these two direction (left-right? up-down? north-south?) we are talking about in regards to bidirectional reflectance distribution function (BRDF), bidirectional scattering distribution function (BSDF), or bidirectional transmittance distribution function (BTDF) in computer graphics?
In general the two directions in BxDF are incoming $\omega_i$ and outgoing $\omega_o$ radiance directions, often defined in spherical coordinates $[\theta, \phi]$ or as a 3D unit vector in Cartesian coordinates $[x, y, z]$. The BxDF $f(\omega_i, \omega_o)$ defines how much of the incident radiance $L_i$ from direction $\omega_i$ hitting the surface is scattered towards direction $\omega_o$. What $\omega_i$ and $\omega_o$ exactly represent depends on the context.

Often $\omega_i$ represents the direction of a light source, but it could also be the direction of a surfel reflecting light or some other light emitting/scattering element in space. $\omega_o$ often represent the direction of the viewpoint (camera/eye), but it could as well be the direction of a surfel or other point in space whose incident radiance we are interested in calculating.

While $\omega_o$ is valid for normal oriented hemisphere above the surface, there are differences in valid $\omega_i$ domain between BxDF's. Because BRDF defines the amount of reflected light $\omega_i$ is valid for the normal oriented hemisphere above the surface. On the other hand, BTDF defines the amount of light transmitted through the surface so $\omega_i$ is valid for the normal oriented hemisphere below the surface. BSDF is the sum of the two so $\omega_i$ is valid for the entire sphere.