求解二维和三维法向量的方法

二维曲线法向量的求解示例：

Here's an example using an analytic curve of y = x^2

   
   
   
   

    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      

       
       
       
       x = 
       
       
       
       0:
       
       
       
       0.1:
       
       
       
       1;
      
      
      
      
     
     
     
     
    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      

       
       
       
       y = x.*x;
      
      
      
      
     
     
     
     
    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      

       
       
       
       dy = gradient(y);
      
      
      
      
     
     
     
     
    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      

       
       
       
       dx = gradient(x);
      
      
      
      
     
     
     
     
    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      

       
       
       
       quiver(x,y,-dy,dx)
      
      
      
      
     
     
     
     
    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      

       
       
       
       hold 
       
       
       
       on; plot( x, y)
      
      
      
      
     
     
     
     
   
   
   
   
   
   
   
   

which gives:

三维三角面片顶点法向量的求解：顶点相邻各个三角面片单位法向量的叠加。

http://www.lighthouse3d.com/opengl/terrain/index.php3?normals

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:

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.

The vertex normal v is computed as the normalised sum of all vij vectors:

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       	v = normalised(sum(v12, v23, v34, v41))
      
      
      
      
     
     
     
     
    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      
	
      
      
      
      
     
     
     
     
    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      
	
       
       
       
       where
      
      
      
      
     
     
     
     
    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      
	
      
      
      
      
     
     
     
     
    
    
    
    

     
     
     
     

      
      
      
      

     
     
     
     
     
     
     
     

      
      
      
      

       
       
       
       		vij = normalised(vi x vj) 
       
       
       
       // normalised cross product