Based on the answer from kvark, I would like to add more thoughts.
If you are in need of an orthonormalized tangent space matrix you have to do some work any way.
Even if you add tangent and binormal attributes, they will be interpolated during the shader stages
and at the end they are neither normalized nor they are normal to each another.
Let's assume that we have a normalized normalvector n, and we have the tangent t and the binormalb or we can calculate them from the derivations as follows:
// derivations of the fragment position
vec3 pos_dx = dFdx( fragPos );
vec3 pos_dy = dFdy( fragPos );
// derivations of the texture coordinate
vec2 texC_dx = dFdx( texCoord );
vec2 texC_dy = dFdy( texCoord );
// tangent vector and binormal vector
vec3 t = texC_dy.y * pos_dx - texC_dx.y * pos_dy;
vec3 b = texC_dx.x * pos_dy - texC_dy.x * pos_dx;
Of course an orthonormalized tangent space matrix can be calcualted by using the cross product,
but this would only work for right-hand systems. If a matrix was mirrored (left-hand system) it will turn to a right hand system:
t = cross( cross( n, t ), t ); // orthonormalization of the tangent vector
b = cross( n, t ); // orthonormalization of the binormal vector
// may invert the binormal vector
mat3 tbn = mat3( normalize(t), normalize(b), n );
In the code snippet above the binormal vector is reversed if the tangent space is a left-handed system.
To avoid this, the hard way must be gone:
t = cross( cross( n, t ), t ); // orthonormalization of the tangent vector
b = cross( b, cross( b, n ) ); // orthonormalization of the binormal vectors to the normal vector
b = cross( cross( t, b ), t ); // orthonormalization of the binormal vectors to the tangent vector
mat3 tbn = mat3( normalize(t), normalize(b), n );
A common way to orthogonalize any matrix is the Gram–Schmidt process:
t = t - n * dot( t, n ); // orthonormalization ot the tangent vectors
b = b - n * dot( b, n ); // orthonormalization of the binormal vectors to the normal vector
b = b - t * dot( b, t ); // orthonormalization of the binormal vectors to the tangent vector
mat3 tbn = mat3( normalize(t), normalize(b), n );
Another possibility is to use the determinant of the 2*2 matrix, which results from the derivations of the texture coordinates texC_dx, texC_dy, to take the direction of the binormal vector into account. The idea is that the determinant of a orthogonal matrix is 1 and the determined one of a orthogonal mirror matrix -1.
The determinant can eihter be calcualted by the GLSL function determinant( mat2( texC_dx, texC_dy )
or it can be calcualated by it formula texC_dx.x * texC_dy.y - texC_dy.x * texC_dx.y.
For the calculation of the orthonormalized tangent space matrix, the binormal vector is no longer required and the calculation of the unit vector
(normalize) of the binormal vector can be evaded.
float texDet = texC_dx.x * texC_dy.y - texC_dy.x * texC_dx.y;
vec3 t = texC_dy.y * pos_dx - texC_dx.y * pos_dy;
t = normalize( t - n * dot( t, n ) );
vec3 b = cross( n, t ); // b is normlized because n and t are orthonormalized unit vectors
mat3 tbn = mat3( t, sign( texDet ) * b, n ); // take in account the direction of the binormal vector
