《Unity着色器圣经》5.0.4 | 叉乘

目录索引

译文

叉乘（又称向量积）也是一种向量运算，与点乘不同的是，叉乘返回的是一个与输入向量组成的平面垂直的向量。

为了充分理解叉乘的概念，让我们假设现在有向量 a 和 向量b，它们的坐标如下所示：

• 向量 a (1, 0, 0)
• 向量 b (0, 1, 0)

在上面的例子中，叉乘得到了一个具有新坐标的向量“c”。

要想理解交叉积的操作，我们必须关注下面的操作：

|| a × b || = || a || || b || Sin θ

叉乘结果所得到的向量大小与正弦函数有关。同时，我们可以通过这向量 a 和向量 b 之间的行列式矩阵计算出交叉积。

• 向量 a (1, 0, 0)
• 向量 b (0, 1, 0)

我们根据向量 a 和向量 b 的坐标计算第三个向量：

vector c = x [(ay * bz) – (az * by)] y [(az * bx) – (ax * bz)] z [(ax * by) – (ay * bx)]

通过替换具体的值，可以得到：

vector c = x [(0 * 0) – (0 * 1)] y [(0 * 1) – (1 * 0)] z [(1 * 1) – (0 * 0)]
vector c = x [(0 – 0)] y [(0 – 0)] z [(1 – 0)]

• 向量 c = (0, 0, 1)

最后，我们可以发现得到的新向量“c”与向量 a 、向量 b 垂直。

在本书之后的章节中，我们将使用叉乘函数计算法线贴图中副切线的值。

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原文对照

The cross product (also known as the vector product) is an operation that, unlike the dot product, returns a three-dimensional vector that is perpendicular to its arguments.

To understand this concept, we will take two vectors: a and b, and position them in space as follows.

• vector a (1, 0, 0)
• vector b (0, 1, 0)

In the example above, the cross product generates a third vector named “c” with new space coordinates.

To understand the operation of the cross product, we must pay attention to the next operation.

|| a × b || = || a || || b || Sin θ

The magnitude of the resulting vector will be related to the function of sin.

Taking into consideration the vectors a and b mentioned above, we can calculate the cross product from a determinant matrix between both vectors.

• vector a (1, 0, 0)
• vector b (0, 1, 0)

We calculate the third vector from the a and b values.

vector c = x [(ay * bz) – (az * by)] y [(az * bx) – (ax * bz)] z [(ax * by) – (ay * bx)]

By replacing the values, we get.

vector c = x [(0 * 0) – (0 * 1)] y [(0 * 1) – (1 * 0)] z [(1 * 1) – (0 * 0)]
vector c = x [(0 – 0)] y [(0 – 0)] z [(1 – 0)]

• vector c = (0, 0, 1)

In conclusion, the resulting vector is perpendicular to its arguments.

Later, we will use the cross product function to calculate the value of a binormal in a normals map.