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Rotation about an arbitrary axis

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Hello everyone, I am a beginner to graphics programming using opengl, so my question probably sounds very simple. I real app. if someone can help me on this. I never took any linear algebra so, all this matrix stuffs are getting me VERY confused. Basicly I am writing a basic fight simulator, and I am trying to get row, yaw, and pitch to work. I want my camera to rotate about its own local axis, not world axis. For example, if my camera/plane first does a 10 degree row, the current transformation matrix will become something like this, +0.984808 -0.173648 +0.000000 +0.000000 +0.173648 +0.984808 +0.000000 +0.000000 +0.000000 +0.000000 +1.000000 +0.000000 +0.000000 +0.000000 +0.000000 +1.000000 Now if I want to do a pitch, I need rotate around the [+0.984808 -0.173648 +0.000000] vector not [1, 0, 0]. So the current transformation matrix becomes, +0.984808 -0.173648 +0.000000 +0.000000 (remains the same) +up_vec_x +up_vec_y +up_vec_z +0.000000 +fwd_vec_x +fwd_vec_y +fwd_vec_z +0.000000 +0.000000 +0.000000 +0.000000 +1.000000 How do I compute the unknown values? I really hope what I am thinking is the correct way to do this. Also, I know I am asking alot here, but an example code will really help me to understand faster. Thank you very much.

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You'll want to look here to do it the transformation-matrix way.

If you feel comfortable with it, a more efficient way to do it is to use quaternions, the four-dimensional generalizations of complex numbers.

Given angle θ in radians and unit vector u = ai + bj + ck, define:

q0 = cos(θ/2)
q1 = a*sin(θ/2)
q2 = b*sin(θ/2)
q3 = c*sin(θ/2)

Then the rotation matrix to apply to u is:

row 1:
col 0 --> (q0² + q1² - q2² - q3²),
col 1 --> 2(q1q2 - q0q3),
col 2 --> 2(q1q3 + q0q2)

row 2:
col 0 --> 2(q2q1 + q0q3),
col 1 --> (q0² - q1² + q2² - q3²),
col 2 --> 2(q2q3 - q0q1)

row 3:
col 0: --> 2(q3q1 - q0q2),
col 1: --> 2(q3q2 + q0q1)
col 2: --> (q0² - q1² - q2² + q3²)

You can read more about an explanation of the quaternion method here, and more on quaternions in general here.

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