[TOC]

看到一段代码，顿时就呆了

    if (angle != angle)
    {
        return XMQuaternionIdentity();
    }

这样写的用意到底是啥呢？

源代码

代码很好理解，就是求两个向量的四元数表示的旋转角度

DirectX::XMVECTOR SimStarMan::QuaternionFromToRotation(const XMVECTOR& from, const XMVECTOR& to)
{
    XMVECTOR quatRet;
    XMVECTOR axis, fromN, toN;
    float angle;
    fromN = XMVector3Normalize(from);
    toN = XMVector3Normalize(to);
    const XMVECTOR epsilon = XMLoadFloat3(&XMFLOAT3(0.00001f, 0.00001f, 0.00001f));

    if (XMVector3NearEqual(fromN, toN, epsilon))
    {
        return XMQuaternionIdentity();
    }
    axis = XMVector3Cross(fromN, toN);
    axis = XMVector3Normalize(axis);
    angle = acosf(XMVectorGetX(XMVector3Dot(fromN, toN)));
    if (angle != angle)
    {
        return XMQuaternionIdentity();
    }
    quatRet = XMQuaternionRotationAxis(axis, angle);
    quatRet = XMQuaternionNormalize(quatRet);
    return quatRet;
}

探究

NAN

参考[2]中解释了，说IEEE标准规定，NaN和任何数比较都是false，所以只有当angle为NaN的时候， angle != angle才为true

float a = acos(2);
cout << "a = " << a << endl;    // a = nan
cout << (a != a) << endl;         // 1

向量可能有问题

比如某个向量是无限长，正如XMVector3Normalize文档提到的一样[3]

For a vector of length 0, this function returns a zero vector. For a vector with infinite length, it returns a vector of QNaN.

acosf会出现Nan的情况

如果输入范围有问题，那么结果就会出现NaN，第二条提到了[4]

If no errors occur, the arc cosine of arg (arccos(arg)) in the range [0 ; π], is returned.

If a domain error occurs, an implementation-defined value is returned (NaN where supported).

If a range error occurs due to underflow, the correct result (after rounding) is returned.

如何写更优雅？

参考[2]中有个评论是这么回答的：

This answer should be updated since std::isnan is now part of the C++11 standard and support has spread out. std::isnan was implemented in Visual Studio starting with Visual Studio 2013.

所以，上面的写法是不是写成下面这样，会更加清晰明了：

// old
angle ! = angle
// new
std::isnan(angle)

至此，这个问题算是清楚了，那么NaN到底是如何定义的呢？

NaN定义

参考[5][6]

IEEE 754 floating point numbers can represent positive or negative infinity, and NaN (not a number).

小结：

NaN在运算中得结果始终都是NaN，不像infinity有可能算出来一个正常值，比如 4/∞ = 0
NaN的浮点数IEEE表示是一个范围，节码都是1，所以尾数表示的范围都是NaN

Positive infinity is represented by the bit pattern X'7F80 0000'.
Negative infinity is represented by the bit pattern X'FF80 0000'.
A signaling NaN (NANS) is represented by any bit pattern between X'7F80 0001' and X'7FBF FFFF' or between X'FF80 0001' and X'FFBF FFFF'.
A quiet NaN (NANQ) is represented by any bit pattern between X'7FC0 0000' and X'7FFF FFFF' or between X'FFC0 0000' and X'FFFF FFFF'.

     31
     |
     | 30    23 22                    0
     | |      | |                     |
-----+-+------+-+---------------------+
qnan 0 11111111 10000000000000000000000
snan 0 11111111 01000000000000000000000
 inf 0 11111111 00000000000000000000000
-inf 1 11111111 00000000000000000000000
-----+-+------+-+---------------------+
     | |      | |                     |
     | +------+ +---------------------+
     |    |               |
     |    v               v
     | exponent        fraction
     |
     v
     sign