从泰特-布赖恩角的四元数

Question

我正在用SharpDX编写照相机，并在四元数的帮助下旋转它。

• 摄像机旋转设置为俯仰（X旋转），偏航（Y旋转）和横滚（Z旋转），即所谓的“ Tiat-Bryan”角（这些是not欧拉角，将具有XYX旋转，而不是XYZ）。
• 我正在使用左手坐标系：X +向右，Y +向上，Z +向下屏幕。
• 如果我绕X旋转（“俯仰”），则正值会使相机朝下看。如果我绕Y旋转（“偏航”），则正值会使相机向左看。如果我绕Z旋转（“滚动”），相机将顺时针旋转。

我已经能够从四元数s中获得俯仰，偏航和侧倾。以下C＃代码：

public static class QuaternionExtensions
{        
    public static Vector3 GetPitchYawRoll(this Quaternion q)
    {
        return new Vector3(q.GetPitch(), q.GetYaw(), q.GetRoll());
    }

    public static float GetPitch(this Quaternion q)
    {
        return q.GetK().GetPitch();
    }

    public static float GetYaw(this Quaternion q)
    {
        return q.GetK().GetYaw();
    }

    public static float GetRoll(this Quaternion q)
    {
        // This is M12 * M22 of rotation matrix
        float xx = q.X * q.X;
        float xy = q.X * q.Y;
        float zz = q.Z * q.Z;
        float wz = q.W * q.Z;
        return (float)Math.Atan2(2f * (xy - wz), 1f - 2f * (xx + zz));
    }

    public static Vector3 GetK(this Quaternion q)
    {
        float xz = q.X * q.Z;
        float wy = q.W * q.Y;
        float yz = q.Y * q.Z;
        float wx = q.W * q.X;
        float xx = q.X * q.X;
        float yy = q.Y * q.Y;
        return new Vector3(
            2f * (xz - wy),
            2f * (yz + wx),
            1f - 2f * (xx + yy));
    }

}

public static class Vector3Extensions
{
    public static float GetPitch(this Vector3 v)
    {
        return (float)-Math.Atan2(v.Y, Math.Sqrt(v.X * v.X + v.Z * v.Z));
    }

    public static float GetYaw(this Vector3 v)
    {
        return (float)-Math.Atan2(v.X, v.Z);
    }
}

我不知道反过来。如何通过指定俯仰，偏航和横摇来获取四元数（例如，Vector3从GetPitchYawRoll返回的内容）？

Answer 1

来自BlitzBasic.com论坛的Kryzon在这里帮助了我（请参阅http://blitzbasic.com/Community/post.php?topic=102458&post=1224801中的帖子：]

internal static class QuaternionExtensions
{
    internal static Vector3 GetYawPitchRollVector(this Quaternion q)
    {
        return new Vector3(q.GetYaw(), q.GetPitch(), q.GetRoll());
    }

    private static float GetYaw(this Quaternion q)
    {
        float x2 = q.X * q.X;
        float y2 = q.Y * q.Y;
        return (float)Math.Atan2(2f * q.Y * q.W - 2f * q.Z * q.X, 1f - 2f * y2 - 2f * x2);
    }

    private static float GetPitch(this Quaternion q)
    {
        return (float)-Math.Asin(2f * q.Z * q.Y + 2f * q.X * q.W);
    }

    private static float GetRoll(this Quaternion q)
    {
        float x2 = q.X * q.X;
        float z2 = q.Z * q.Z;
        return (float)-Math.Atan2(2f * q.Z * q.W - 2f * q.Y * q.X, 1f - 2f * z2 - 2f * x2);
    }
}

internal static class Vector3Extensions
{
    internal static Quaternion GetYawPitchRollQuaternion(this Vector3 v)
    {
        float c1 = (float)Math.Cos(-v.Z / 2.0);
        float c2 = (float)Math.Cos(-v.Y / 2.0);
        float c3 = (float)Math.Cos( v.X / 2.0);
        float c1c2 = c1 * c2;
        float s1 = (float)Math.Sin(-v.Z / 2.0);
        float s2 = (float)Math.Sin(-v.Y / 2.0);
        float s3 = (float)Math.Sin( v.X / 2.0);
        float s1s2 = s1 * s2;

        return new Quaternion(
            c1 * s2 * c3 - s1 * c2 * s3,
            c1c2 * s3 + s1s2 * c3,
            s1 * c2 * c3 + c1 * s2 * s3,
            c1c2 * c3 - s1s2 * s3);
    }
}

Answer 2

通常被称为欧拉角，

http://en.wikipedia.org/wiki/Euler_angles

可以在https://github.com/erich666/GraphicsGems/blob/master/gemsiv/euler_angle/EulerAngles.c找到所有转换的变体

最初由Ken Shoemake撰写，1993