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Comparison of 2 different quaternion axes

I am trying to compare the Z-axis and the X-axis of two different quaternions in a way that would give me the Euler angles about the X and Y axes to line up the two different axes. In my program, the first quaternion (whose X-axis we are trying to line up) only rotates about the z and y axes and is the one being rotated first. Whereas the second quaternion (whose Z-axis we are trying to line up) only rotates about the x and y axis and is the one being rotated to match the other quaternion. Currently, I've been able to deduce the direct distance to line up the axes though am struggling to break the angles into pitch and yaw.

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