3D Translations relative to 3D Rotations

Question

I'm trying to program camera movement to be relative to camera rotation. (Forward is always forward, regardless of pitch, yaw, and roll) I want to be able to move forward, backward, left, right, up, and down.

I do not want to use a matrix.

I want to use sin and cos from the standard math library.

The camera rotates on all 3 axis. The rotation order is z(roll), y(pitch), x(yaw). When all rotations are 0, positive z is forward, positive y is down, and positive x is left.

So far I've gotten forward and backward movement to work with:

velZ = speed * cos(rotX) * cos(rotY);
velY = speed * sin(rotY);
velX = speed * sin(-rotX) * cos(rotY);

If somebody knows how to do this or knows where I can find information on this, it would be greatly appreciated.

Answer 1

In neutral position you have defined forward to be the positive Z vector (0, 0, 1). There are two vectors perpendicular to that vector (if we ignore sign), up (0, 1, 0) and left (1, 0, 0).

The easiest thing would be to create all three vectors and to apply a matrix transformation to find the left, up, and forward vector in 'camera space'.

Matrix transform = Matrix.CreateFromYawPitchRoll(y, p r);
Vector3 forward = Vector3.Transform(transform, new Vector3(0, 0, 1));
Vector3 left = Vector3.Transform(transform, new Vector3(1, 0, 0));
Vector3 up = Vector3.Transform(transform, new Vector3(0, 1, 0));

You do not necessarily need a matrix. You can also find the left and up matrix by using the cross product. (See here Unity's documentation on it, but the mathematical principals apply in general).

Once you have all three vectors you can strafe and move up by simply adding the correct vector (multiplied by movement speed) to the camera position (and look at) vector(s). For example:

public void MoveLeft(float speed)
{
    camera.Position = camera.Position + (left * speed);
}