Terrain Tutorial
地形教程
Computing Normals
法線的計算
To apply lighting to a terrain, either using OpenGL lights, or simulating them, it is necessary to first compute normals. A normal is a vector that defines how a surface responds to lighting, i.e. how it is lit. The amount of light reflected by a surface is proportional to the angle between the lights direction and the normal. The smaller the angle the brighter the surface will look.
為了給地形進行光照,任意一個OpenGL的光照,或者模擬它們, 那么我們必須要首先計算法線. 法線是一個向量定義了表面對光照的響應.例如,如何去照亮它.表面的反射光量是與光線方向與法線方向的夾角成正比. 夾角越小表面就會看起來越亮.
Normals in OpenGL can be defined per face or per vertex. If defining a normal per face then the normal is commonly defined as a vector which is perpendicular to the surface. In order to find a perpendicular vector to a face, two vectors coplanar with the face are needed. Afterwards the cross product will provide the normal vector, i.e. a perpendicular vector to the face.
OpenGL里面可以定義面法線和頂點法線. 如果定義了面法線,那么這個法線向量一般都要正交于這個表面. 為了去找到面的正交向量, 需要兩個共面向量. 然后它們叉乘就會產生法向量, 一個正交于面的向量.
So the first step is to compute two vectors coplanar to a face. Assuming the the faces are triangles defined by points t1,t2,t3, then two possible vectors are:
所以首先來計算面的兩個共面的向量.假設面是三角形用t1,t2,t3定義, 然后兩個可能的向量是:
v1 = t2 - t1
v2 = t3 - t1
With the two vectors, v1 and v2, it is now possible to compute the cross product between them to find a perpendicular vector to the face.
有了兩個向量v1和v2, 現在我們就可以計算叉乘,來找到正交于面的一個向量.
The equations bellow show the necessary steps to compute a normal vector v. The required opeartion is called cross product, and it is represented by "x".
下面的等式展示了計算法線v的必要的一些步驟. 這個需要的運算叫做叉乘, 我們把它表示為"X"
v = v1 x v2
v = [vx,vy,vz] where,
vx = v1y * v2z - v1z * v2y
vy = v1z * v2x - v1x * v2z
vz = v1x * v2y - v1y * v2x
Another necessary step to obtain proper lighting is to normalise the vector, i.e. make it unit length. OpenGL takes into consideration the length of the normal vector when computing lighting. Normalisation implies first computing the lenght of the vector, and then dividing each component by the vectors length.
另一個必要的步驟獲得正確的光照是歸一化這個向量, 就是讓它是單位長度. OpenGL在計算光照的時候需要考慮歸一化的法向量.歸一化的過程意味著第一步首先計算向量的長度,然后向量的每個部分除以此長度.
The length of a vector is computed as:
向量的長度是以下公式計算來:
Therefore the normalized vector nv is computed as:
因此,歸一化的向量nv計算過程如下:
nv = [nvx,nvy,nvz] where,
nvx = vx / l
nvy = vy / l
nvz = vz / l
The main problem with assigning a normal per face is that the terrain looks faceted, i.e. the brightness of each face is constant, and there is a clear difference between faces with differen orientations. In order to get a smoother look normals should be computed per vertex, and not per face. When computing normals per vertex it is necessary to take into account the faces that share the vertex. So for instance if using quads, each vertex (excluding the corner and border vertices), is shared by four polygons. The normal at a vertex should be computed as the normalised sum of all the unit length normals for each face the vertex shares. Consider the following image:
為地形使用面法線最大的問題就是它看起來一塊一塊的， 因為每個面的亮度都是恒定的，并且每個方向的面上的亮度都有明顯的區別。為了看起來更加的光滑，我們必須為每個頂點計算法線，而不是每個面計算法線。當計算頂點法線的時候，我們有必要考慮到此頂點共享的所有面，所以我們用方形舉例，每個頂點（排除掉角落和邊緣的頂點）被四個多邊形共享。這個頂點的法線就應該是所有共享面法線的和在歸一化的結果。看看下面的圖：
In the above image, v represents the normal at the center vertex. Each vij represents a normal for each face that shares the center vertex. So for instance v12 is the unit lenght normal for the bottom right face.
在上面的圖里，v代表了中心頂點的法線。每個vij表示每個共享面的法線。所以例如v12是右下方面的單位長度的法線。
The vertex normal v is computed as the normalised sum of all vij vectors:
頂點法線v被計算作為所有vij向量歸一化后的和。
v = normalised(sum(v12, v23, v34, v41))
where
vij = normalised(vi x vj) // normalised cross product
It is also posible to consider the eight neighbour vertices, instead of only four. This latter option will probably look smoother in the general case.
也可以考慮8個相臨的頂點。這個是后來的選項比一般的做法看起來更加光滑。
Note that when computing the normals a scale is assumed. If the application has performed non-uniform scaling the normals will no longer be correct. If scaling the heights is required use the function terrainScale provided in the terrain library. This function recomputes the normals. If the grid needs scaling then use the function terrainDim to enlarge the terrain
注意當計算法線時假設了一個縮放。如果程序使用了一個非歸整的縮放，那么法線將不再正確。如果縮放了高度，那么就需要使用函數terrainScale來調整。這個函數重新計算了法線。如果網格需要縮放，那么就使用terrainDim函數來擴大地形。