convert orientation vec3 to a rotation matrix

Question

I've got a normalized vec3 that represents an orientation.

Each frame of animation, an object's orientation changes slightly, so I add a delta vector to the orientation vector and then normalize to find the new orientation.

I'd like to convert the vec3 that represents an orientation into a rotation matrix that I can use to orient my object.

If it helps, my object is a cone, and I'd like to rotate it about the pointy end, not from its center :)

PS I know I should use quaternions because of the gimbal lock problem. If someone can explain quats too, that'd be great :)

Answer 1

Your problem is under-constrained, so there are a lot of possible solutions.

My suggestion is to see your vec3 as the result of rotating vector [1 0 0] by φ around axis Y then by θ around axis Z. This is the latitude/longitude notation. The corresponding transformation matrix is:

|cosθ cosφ   -sinθ   -cosθ sinφ|
|sinθ cosφ    cosθ   -sinθ sinφ|
|   sinφ       0         cosφ  |

See the first column? That’s your vec3, since it’s the image of [1 0 0]. So the good thing is that we already know a lot of the matrix values.

The following code computes the remaining values:

mat3 rotation_matrix(vec3 v)
{
    /* Find cosφ and sinφ */
    float c1 = sqrt(v.x * v.x + v.y * v.y);
    float s1 = v.z;
    /* Find cosθ and sinθ; if gimbal lock, choose (1,0) arbitrarily */
    float c2 = c1 ? v1.x / c1 : 1.0;
    float s2 = c1 ? v1.y / c1 : 0.0;

    return mat3(v.x, -s2, -s1*c2,
                v.y,  c2, -s1*s2,
                v.z,   0,     c1);
}

Be careful with the last line, it may need a transposition depending on how your matrix class works.

Answer 2

A single vector only represents a direction. It sounds like gimbal lock will not be a problem in your case.

First you need to select an up vector to really define an orientation instead of a direction. You can start with the world's up vector and than use the last frames up vector from there on. Now you can calculate a side vector using the cross product and correct the up vector to ensure you have an orthonormal basis. To get the rotation matrix you just put these three vectors into a matrix's rows or columns depending on your coordinate system and used matrix convention.

Vec3 forward = your_rotation;
Vec3 up;
Vec3 side;

side = cross(forward, last_up);
if( side.GetLengthSq() < 0.001f )
{
  side = cross(forward, world_up);
  if( side.GetLengthSq() < 0.001f )
    side = cross(forward, world_side);
}

side.Normalize();
up = cross( forward, side );

// example, depends solely on the matrix format and coordinate space used
Matrix3 rotation( forward, side, up );

The easier way is to use a lookat function which does exactly the same and is provided with most engines and render APIs.