有效的方式来反映四元数旋转的效果呢？ [英] Efficient way to mirror effect of quaternion rotation?

问题描述

四元数重新present旋转 - 它们不包含有关缩放或镜像信息。然而，它仍然是可能的，以镜像旋转的效果。

Quaternions represent rotations - they don't include information about scaling or mirroring. However it is still possible to mirror the effect of a rotation.

考虑在xy平面上的镜像（我们也可以把它沿z轴的镜像）。绕x轴的镜像在xy平面上的旋转将被否定。同样地与围绕y轴的旋转。然而，绕z轴的旋转将保持不变。

Consider a mirroring on the x-y-plane (we can also call it a mirroring along the z-axis). A rotation around the x-axis mirrored on the x-y-plane would be negated. Likewise with a rotation around the y axis. However, a rotation around the z-axis would be left unchanged.

又如：绕轴线90度旋转（1,1,1）反映在xy平面会给-90°左右旋转（1,1，-1）。为了帮助直觉，如果你能想象轴的描绘和一个圆形箭头指示旋转，然后镜像的可视化指明了新的转动应。

Another example: 90º rotation around axis (1,1,1) mirrored in the x-y plane would give -90º rotation around (1,1,-1). To aid the intuition, if you can visualize a depiction of the axis and a circular arrow indicating the rotation, then mirroring that visualization indicates what the new rotation should be.

我已经找到一种方法来计算这个镜像旋转，像这样的：

I have found a way to calculate this mirroring of the rotation, like this:

• 获取四元数的角度轴重新presentation。
• 对于每个轴x，y和z的。
  • 如果换算为负（镜像）沿轴：
    • 在否定双方的角度和轴。
    • Get the angle-axis representation of the quaternion.
    • For each of the axes x, y, and z.
      • If the scaling is negative (mirrored) along that axis:
        • Negate both angle and axis.

        这仅支持沿主坐标轴，X，Y和Z镜像，因为这是我所需要的。它的工作原理为，虽然任意旋转。

        This only supports mirroring along the primary axes, x, y, and z, since that's all I need. It works for arbitrary rotations though.

        不过，从四元数角度轴的转换和从角度轴回四元数是昂贵的。我不知道是否有一种方法来直接进行转换的四元数本身，而是我的COM $ P $四元数的数学phension不足以取得任何自己。

        However, the conversions from quaternion to angle-axis and back from angle-axis to quaternion are expensive. I'm wondering if there's a way to do the conversion directly on the quaternion itself, but my comprehension of quaternion math is not sufficient to get anywhere myself.

        （发布于计算器，而不是数学有关的论坛由于计算有效方法的重要性。）

        (Posted on StackOverflow rather than math-related forums due to the importance of a computationally efficient method.)

        推荐答案

        我做了一些进一步的分析，它会出现一个四元数的影响（W，X，Y，Z）可以有它反映这样的效果：

        I did some further analysis, and it appears the effect of a quaternion (w, x, y, z) can have it's effect mirrored like this:

        • 沿X轴旋转翻转四元数y和z元素的镜面效果。
        • 在沿Y轴旋转的镜面效果通过翻转四元数的X和Z元素。
        • 在沿Z轴的旋转的镜面效果翻转x和四元数ÿ元素。

        四元数的W元素永远需要被感动。

        The w element of the quaternion never needs to be touched.

        不幸的是我还是不明白，四元数足够好，能够解释为什么这个工作，但我得出它的实现转换，并从轴的角度格式，并实施该解决方案后，它的工作原理一样好我原来在它我已经执行了所有的测试。

        Unfortunately I still don't understand quaternions well enough to be able to explain why this works, but I derived it from implementations of converting to and from axis-angle format, and after implementing this solution, it works just as well as my original one in all tests of it I have performed.

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