glsl The gl_NormalMatrix 方向向量的转换

GLSL Tutorial

The gl_NormalMatrix

The gl_NormalMatrix is present in many vertex shaders. In here some light is shed on what is this matrix and what is it for. This section was inspired by the excellent book by Eric Lengyel "Mathematics for 3D Game Programming and Computer Graphics".

Many computations are done in eye space. This has to do with the fact that lighting needs to be performed in this space, otherwise eye position dependent effects, such as specular lights would be harder to implement. //很多计算都是在眼坐标中进行的

Hence we need a way to transform the normal into eye space. To transform a vertex to eye space we can write: //因此需要把顶点转找到眼坐标中

	vertexEyeSpace = gl_ModelViewMatrix * gl_Vertex;

So why can't we just do the same with a normal vector? First a normal is a vector of 3 floats and the modelview matrix is 4x4. This could be easily overcome with the following code:

//但为什么不能直接转换方向向量呢?因为方向向量只有三个分量,但我们可以用下面的代码解决

	normalEyeSpace = vec3(gl_ModelViewMatrix * vec4(gl_Normal,0.0));

So, gl_NormalMatrix is just a shortcut to simplify code writing? No, not really. The above line of code will work in some circunstances but not all. //上面的代码并不是在所有的情况下都完全正确,我们从下面看一下存在的潜在问题

Lets have a look at a potential problem:

In the above figure we see a triangle, with a normal and a tangent vectors. The following figure shows what happens when the modelview matrix contains a non-uniform scale.
上图显示了一个三角形,并且法线垂直于平面,下图显示法线经过非均衡缩放后,法线不再垂直平面了

Note: if the scale was uniform, then the direction of the normal would have been preserved, The length would have been affected but this can be easily fixed with a normalization.
//如果x,y,z三个方向缩放比例相同的话,虽然法线向量的长度改变了,但是方向不变,这种情况是可以用单位化向量的方法解决

In the above figure the modelview matrix was applied to all the vertices as well as to the normal and the result is clearly wrong: the normal is no longer perpendicular to the surface.

So now we know that we can't apply the modelview in all cases to transform the normal vector. The question is then, what matrix should we apply?

We know that, prior to the matrix transformation T.N = 0, since the vectors are by definition perpendicular. We also know that after the transformation N'.T' must remain equal to zero, since they must remain perpendicular to each other. Let's assume that the matrix G is the correct matrix to transform the normal vector. T can be multiplied safely by the upper left 3x3 submatrix of the modelview (T is a vector, hence the w component is zero). This is because T can be computed as the difference between two vertices, therefore the same matrix that is used to transform the vertices can be used to transform T. Hence the following equation: 转换后的两个向量的点乘应该为0,矩阵G是未知数,是假设的正确的矩阵,M就是顶点的转换矩阵,也就是模型视图矩阵,而向量N和T是转换前的平面的法向量和切向量.

 

The dot product can be transformed into a product of vectors, therefore:
由于GN相乘的结果是向量,因此GN等于GN的转置(一个向量和它的转置相同)

Note that the transpose of the first vector must be considered since this is required to multiply the vectors. We also know that the transpose of a multiplication is the multiplication of the transposes, hence:
GN乘积的转置等于它们转置的乘积
 

We started by stating that the dot product between N and T was zero, so if the following equation is true then we are on the right track.
由于上式结果为0,因此下式成立(可用反证法,如果下式结果不为单位阵,则上式必不为0向量)

Applying a little algebra yieds
下式由上式进行矩阵运算得到

Therefore the correct matrix to transform the normal is the transpose of the inverse of the M matrix. OpenGL computes this for us in the gl_NormalMatrix.

In the beginning of this section it was stated that using the modelview matrix would work in some cases. Whenever the 3x3 upper left submatrix of the modelview is orthogonal we have:

This is because with an orthogonal matrix, the transpose is the same as the inverse. So what is an orthogonal matrix? An orthogonal matrix is a matrix where all columns/rows are unit length, and are mutually perpendicular. This implies that when two vectors are multiplied by such a matrix, the angle between them after transformation by an orthogonal matrix is the same as prior to that transformation. Simply put the transformation preserves the angle relation between vectors, hence normals remain perpendicular to tangents! Furthermore it preserves the length of the vectors as well.

So when can we be sure that M is orthogonal? When we limit our geometric operations to rotations and translations, i.e. when in the OpenGL application we only use glRotate and gl_Translate and not glScale. These operations guarantee that M is orthogonal. Note: gluLookAt also creates an orthogonal matrix!
//--补充
模型视图变化只有平移,旋转的时候才形成正交矩阵,这个时候向量的转换矩阵就是模型视图矩阵,而如果变换包括了平移的时候,则向量的转换矩阵是它的逆的转置