The View-Matrix and Alternative Calculations

Question

I'm working on a radiosity processor in DirectX 9. The process requires that the camera be placed at the center of a mesh face and a 'screenshot' be taken facing 5 different directions...forward...up...down...left...right...

...The problem is that when the mesh face is facing up( look vector: 0, 1, 0 )...a view matrix cannot be determined using standard trigonometry functions:

Matrix4 LookAt( Vector3 eye, Vector3 target, Vector3 up ) 
    {   
        // The "look-at" vector.  
        Vector3 zaxis = normal(target - eye);        
        // The "right" vector.
        Vector3 xaxis = normal(cross(up, zaxis)); 
        // The "up" vector.
        Vector3 yaxis = cross(zaxis, xaxis); 
        // Create a 4x4 orientation matrix from the right, up, and at vectors     
        Matrix4 orientation = 
        {         
            xaxis.x, yaxis.x, zaxis.x, 0,         
            xaxis.y, yaxis.y, zaxis.y, 0,         
            xaxis.z, yaxis.z, zaxis.z, 0,           
            0,       0,       0,     1     
        };           

        // Create a 4x4 translation matrix by negating the eye position.     

        Matrix4 translation = 
        {       
            1,      0,      0,     0,          
            0,      1,      0,     0,            
            0,      0,      1,     0,        
            -eye.x, -eye.y, -eye.z,  1     
        };       
        // Combine the orientation and translation to compute the view matrix     
        return ( translation * orientation ); 
    } 

The above function comes from http://3dgep.com/?p=1700...

...Is there a mathematical approach to this problem?

Answer 1

You can check if cross(up, zaxis) is too close to zero (use an epsilon like 1e-4 or something like that), and switch to an alternative up-vector if so. For instance, if your usual up-vector is (0, 1, 0), you could switch to (1, 0, 0).