如何使矢量绕某一轴旋转

Question

.X3D格式有一个有趣的旋转系统。与大多数包含围绕X、Y和Z轴旋转值的格式不同，.X3D给出了一个归一化方向向量，然后给出了围绕该轴旋转的弧度值。

示例：

The axis to rotate around: 0.000000 0.465391 0.885105 
Rotation around that axis (in radians): 3.141593

我有从弧度到度的转换，但是我需要从这些值中围绕XYZ的旋转值。

Answer 1

我们可以为初等矩阵变换建立一个绕角轴旋转的序列.设轴单位矢量为w，角度为Theta。辅助值：

V = Sqrt(wx*wx + wz*wz)
W = Sqrt(wx*wx + wy*wy + wz*wz)  //1 for unit dir vector
Cos(Alpha) = wz/V
Sin(Alpha) = wx/V
Cos(Beta) = V/W
Sin(Beta) = wy/W

Transformation sequence:
Ry(-Alpha)   //rotation matrix about Y-axis by angle -Alpha
Rx(Beta)
Rz(Theta)
Rx(-Beta)
Ry(Alpha) 

注意，对于is轴是平行于Y的，应该使用关于Y的通常的只旋转矩阵(表示方向符号)，因为V值是零。

有相当复杂的罗德里格斯旋转公式用于计算与旋转相对应的旋转矩阵，该旋转矩阵由单位矢量w所指定的固定轴的θ角来计算。

显式矩阵这里 (怪异格式图片)：

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Answer 2

下面的C++函数围绕A提供的中心旋转一个点，并使用另一个向量(单位)作为轴旋转。下面的代码是由于格伦·默里提供了链接提供的公式而产生的。这是经过测试，并正在完美地按照预期的工作。注:当角度为正，单位向量为{0,0,1}时，它向右旋转，基本旋转在右侧，轴为{x-前进，y-右，z-up}。

void RotateVectorAroundPointAndAxis(Vector3D YourPoint, Vector3D PreferedCenter, Vector3D UnitDirection, float Angle, Vector3D& ReturnVector)
{
    float SinVal = sin(Angle * 0.01745329251);
    float CosVal = cos(Angle * 0.01745329251);
    float OneMinSin = 1.0f - SinVal;
    float OneMinCos = 1.0f - CosVal;   

    UnitDirection = GetUnitVector(UnitDirection, { 0,0,0 });// This Function Gets unit Vector From InputVector - DesiredCenter

    float Temp = (UnitDirection.x * YourPoint.x) + (UnitDirection.y * YourPoint.y) + (UnitDirection.z * YourPoint.z);
    ReturnVector.x = (PreferedCenter.x * (UnitDirection.y * UnitDirection.y)) - (UnitDirection.x * (((-PreferedCenter.y * UnitDirection.y) + (-PreferedCenter.z * UnitDirection.z)) - Temp));
    ReturnVector.y = (PreferedCenter.y * (UnitDirection.x * UnitDirection.x)) - (UnitDirection.y * (((-PreferedCenter.x * UnitDirection.x) + (-PreferedCenter.z * UnitDirection.z)) - Temp));
    ReturnVector.z = (PreferedCenter.z * (UnitDirection.x * UnitDirection.x)) - (UnitDirection.z * (((-PreferedCenter.x * UnitDirection.x) + (-PreferedCenter.y * UnitDirection.y)) - Temp));

    ReturnVector.x = (ReturnVector.x * OneMinCos) + (YourPoint.x * CosVal);
    ReturnVector.y = (ReturnVector.y * OneMinCos) + (YourPoint.y * CosVal);
    ReturnVector.z = (ReturnVector.z * OneMinCos) + (YourPoint.z * CosVal);

    ReturnVector.x += (-(PreferedCenter.z * UnitDirection.y) + (PreferedCenter.y * UnitDirection.z) - (UnitDirection.z * YourPoint.y) + (UnitDirection.y * YourPoint.z)) * SinVal;    
    ReturnVector.y += ( (PreferedCenter.z * UnitDirection.x) - (PreferedCenter.x * UnitDirection.z) + (UnitDirection.z * YourPoint.x) - (UnitDirection.x * YourPoint.z)) * SinVal;
    ReturnVector.z += (-(PreferedCenter.y * UnitDirection.x) + (PreferedCenter.x * UnitDirection.y) - (UnitDirection.y * YourPoint.x) + (UnitDirection.x * YourPoint.y)) * SinVal;    
}