Terrain Tutorial
地形教程
Computing Normals
法線的計(jì)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(jìn)行光照,任意一個(gè)OpenGL的光照,或者模擬它們, 那么我們必須要首先計(jì)算法線. 法線是一個(gè)向量定義了表面對(duì)光照的響應(yīng).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(huì)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(diǎn)法線. 如果定義了面法線,那么這個(gè)法線向量一般都要正交于這個(gè)表面. 為了去找到面的正交向量, 需要兩個(gè)共面向量. 然后它們叉乘就會(huì)產(chǎn)生法向量, 一個(gè)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(jì)算面的兩個(gè)共面的向量.假設(shè)面是三角形用t1,t2,t3定義, 然后兩個(gè)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(gè)向量v1和v2, 現(xiàn)在我們就可以計(jì)算叉乘,來找到正交于面的一個(gè)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(jì)算法線v的必要的一些步驟. 這個(gè)需要的運(yùn)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(gè)必要的步驟獲得正確的光照是歸一化這個(gè)向量, 就是讓它是單位長度. OpenGL在計(jì)算光照的時(shí)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(jì)算向量的長度,然后向量的每個(gè)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(jì)算來:
Therefore the normalized vector nv is computed as:
因此,歸一化的向量nv計(jì)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?yàn)槊總€(gè)面的亮度都是恒定的，并且每個(gè)方向的面上的亮度都有明顯的區(qū)別。為了看起來更加的光滑，我們必須為每個(gè)頂點(diǎn)計(jì)算法線，而不是每個(gè)面計(jì)算法線。當(dāng)計(jì)算頂點(diǎn)法線的時(shí)候，我們有必要考慮到此頂點(diǎn)共享的所有面，所以我們用方形舉例，每個(gè)頂點(diǎn)（排除掉角落和邊緣的頂點(diǎn)）被四個(gè)多邊形共享。這個(gè)頂點(diǎn)的法線就應(yīng)該是所有共享面法線的和在歸一化的結(jié)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(diǎn)的法線。每個(gè)vij表示每個(gè)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(diǎn)法線v被計(jì)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(gè)相臨的頂點(diǎn)。這個(gè)是后來的選項(xiàng)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(dāng)計(jì)算法線時(shí)假設(shè)了一個(gè)縮放。如果程序使用了一個(gè)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(shù)terrainScale來調(diào)整。這個(gè)函數(shù)重新計(jì)算了法線。如果網(wǎng)格需要縮放，那么就使用terrainDim函數(shù)來擴(kuò)大地形。