四元数和三轴 [英] Quaternion and three axes

问题描述

给出一个四元数q和三个3D向量(vx，vy，vz)，它们形成坐标轴，它们可以沿任意方向定向，但彼此垂直，从而形成3d空间.

Given a quaternion q, and three 3D vectors (vx, vy, vz) which form coordinate axes, which can be oriented in arbitrary direction, but are all perpendicular to each other, thus forming a 3d space.

如何检查四元数q是否与某些3D向量(vx，vy，vz)朝相同的方向(或相反的方向)旋转?

How can I check if the quaternion q is rotated to the same direction (or opposite direction) as some of the 3D vectors (vx, vy, vz)?

推荐答案

如果q =(w，x，y，z)，其中w是标量部分"，而qv =(x，y，z)是矢量部分"， 那么您可以计算qv与每个基本向量vx，vy，vz之间的角度 使用点积.

If q = (w,x,y,z), where w is the "scalar part", and qv=(x,y,z) is the "vector part", then you can calculate the angle between qv and each of the basis vectors vx, vy, vz using the dot product.

cos(theta)=(qv点vx)/(| qv | * | vx |)

cos(theta) = (qv dot vx) / ( |qv| * |vx|)

如果cos(theta)为+1，则q的旋转轴平行于该基矢量.

If cos(theta) is +1, the rotation axis of q is parallel to that basis vector.

cos(theta)= -1表示它们是反平行的.

cos(theta) = -1 implies that they are anti-parallel.

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