程序员经验分享-hwhale

人类高质量网格重建算法

2023-10-31

通常对于几何曲面的绘制采用采样描点的方式。比如球的方程为：x2+y2+z2=r2.对于这个方程可以将其改为参数方程。

x=Rcos\alpha cos\beta

y=Rcos\alpha sin\beta

z=Rsin\alpha

然后对\alpha与\beta进行离散化，从而得到一系列的离散点

下面是c++的代码生成一个球

class Sphere:public Mesh
{
public:
	Sphere(vec3 center, double R, bool normal)
	{
		vector<vec3> Vectices;
		vector<vec2>  texcode;
		vector<unsigned int> index;
		//填充索引与坐标
		int uStepNum = 30;
		int vStepNum = 30;
		double ustep = 1 / (double)uStepNum;
		double vstep = 1 / (double)vStepNum;
		double u = 0, v = 0;
		//添加下部的顶点
		for (int i = 0; i < uStepNum; i++)
		{
			vec3 a = center + getPoint(0, 0, R);
			vec3 b = center + getPoint(u, vstep, R);
			vec3 c = center + getPoint(u + ustep, vstep, R);
			u += ustep;
			Vectices.push_back(a);
			texcode.push_back(vec2(0,0));
			index.push_back(Vectices.size() - 1);
			Vectices.push_back(c);
			texcode.push_back(vec2(u + ustep, vstep));
			index.push_back(Vectices.size() - 1);
			Vectices.push_back(b);
			texcode.push_back(vec2(u, vstep));
			index.push_back(Vectices.size() - 1);
		}
		//添加中部的顶点
		u = 0, v = vstep;
		for (int i = 1; i < vStepNum - 1; i++)
		{
			for (int j = 0; j < uStepNum; j++)
			{
				vec3 a = center + getPoint(u, v, R);
				vec3 b = center + getPoint(u + ustep, v, R);
				vec3 c = center + getPoint(u + ustep, v + vstep, R);
				vec3 d = center + getPoint(u, v + vstep, R);
				Vectices.push_back(a);
				texcode.push_back(vec2(u, v));
				index.push_back(Vectices.size() - 1);
				Vectices.push_back(b);
				texcode.push_back(vec2(u + ustep, v));
				index.push_back(Vectices.size() - 1);
				Vectices.push_back(c);
				texcode.push_back(vec2(u + ustep, v + vstep));
				index.push_back(Vectices.size() - 1);
				Vectices.push_back(c);
				texcode.push_back(vec2(u + ustep, v + vstep));
				index.push_back(Vectices.size() - 1);
				Vectices.push_back(d);
				texcode.push_back(vec2(u, v + vstep));
				index.push_back(Vectices.size() - 1);
				Vectices.push_back(a);
				texcode.push_back(vec2(u, v));
				index.push_back(Vectices.size() - 1);
				u += ustep;
			}
			v += vstep;
		}
		//添加上部的顶点
		u = 0;
		for (int i = 0; i < uStepNum; i++)
		{
			vec3  a = center + getPoint(0, 1, R);
			vec3  b = center + getPoint(u, 1 - vstep, R);
			vec3 c = center + getPoint(u + ustep, 1 - vstep, R);
			Vectices.push_back(a);
			texcode.push_back(vec2(0, 1));
			index.push_back(Vectices.size() - 1);
			Vectices.push_back(b);
			texcode.push_back(vec2(u, 1 - vstep));
			index.push_back(Vectices.size() - 1);
			Vectices.push_back(c);
			texcode.push_back(vec2(u + ustep, 1 - vstep));
			index.push_back(Vectices.size() - 1);
			u += ustep;
		}


		for (int i = 0; i < Vectices.size(); i++)
		{
			Vertex temp;
			temp.Position = Vectices[i];
			temp.Normal = vec3(0, 0, 0);
			temp.Bitangent = vec3(0, 0, 0);
			temp.Tangent = vec3(0, 0, 0);
			temp.TexCoords = texcode[i];
			this->vertices.push_back(temp);
		}

		indices = index;
		if (normal)
		{
			calculate_normal();
		}		
		//fill index
		
		this->setupMesh();
		
	}



};

vec3 getPoint(double u, double v,double R)
{
	double x = R*sin(PI*v)*cos(2*PI*u);
	double y = -R * cos(PI*v);
	double z = R * sin(PI*v)*sin(2 * PI*u);
	return vec3(x, y, z);
}

下图是基于上述方法在opengl中渲染出的一些基本图元。

但是对于隐式曲面即f(x,y,z)=0的形式，无法进行参数化，便意味着无法离散化。上述方法就没用了。

隐式曲面的麻烦之处在于其无法参数化，但是有一点好处，在于能轻易的判断出一个点在不在立方体内部。基于这一特性，想到利用移动立方体（marching cube）的方法来重建隐式曲面。移动立方体算法的主要思想是在三维离散数据场中通过线性差值来逼近等值面。在医学图像分割与重建中，我们通过定义一个阈值来确定这个等值面。顶点值有三种情况：高于或等于等值表示在表面的内部，低于等值表示在表面的外部。这样，体元的一个顶点有2中可能的状态，则一个体元（8个顶点）就一共有256种状态。根据旋转、映射不变等特性，体元的状态可以被归纳为以下15种基本构型：

                                        

 

 从而通过每一个立方体顶点的情况，可以判断出每一个立方体内部的网格情况，于是我们便近似出了一个等值面。

下标是256种点的分布对应的三角面片的表格数据。感谢前辈们的整理，这个工作量还是十分巨大的。

int triTable[256][16] =
{{-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 8, 3, 9, 8, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 8, 3, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{9, 2, 10, 0, 2, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{2, 8, 3, 2, 10, 8, 10, 9, 8, -1, -1, -1, -1, -1, -1, -1},
{3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 11, 2, 8, 11, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 9, 0, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 11, 2, 1, 9, 11, 9, 8, 11, -1, -1, -1, -1, -1, -1, -1},
{3, 10, 1, 11, 10, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 10, 1, 0, 8, 10, 8, 11, 10, -1, -1, -1, -1, -1, -1, -1},
{3, 9, 0, 3, 11, 9, 11, 10, 9, -1, -1, -1, -1, -1, -1, -1},
{9, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{4, 3, 0, 7, 3, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 1, 9, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{4, 1, 9, 4, 7, 1, 7, 3, 1, -1, -1, -1, -1, -1, -1, -1},
{1, 2, 10, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{3, 4, 7, 3, 0, 4, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1},
{9, 2, 10, 9, 0, 2, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1},
{2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4, -1, -1, -1, -1},
{8, 4, 7, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{11, 4, 7, 11, 2, 4, 2, 0, 4, -1, -1, -1, -1, -1, -1, -1},
{9, 0, 1, 8, 4, 7, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1},
{4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1, -1, -1, -1, -1},
{3, 10, 1, 3, 11, 10, 7, 8, 4, -1, -1, -1, -1, -1, -1, -1},
{1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4, -1, -1, -1, -1},
{4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3, -1, -1, -1, -1},
{4, 7, 11, 4, 11, 9, 9, 11, 10, -1, -1, -1, -1, -1, -1, -1},
{9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{9, 5, 4, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 5, 4, 1, 5, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{8, 5, 4, 8, 3, 5, 3, 1, 5, -1, -1, -1, -1, -1, -1, -1},
{1, 2, 10, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{3, 0, 8, 1, 2, 10, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1},
{5, 2, 10, 5, 4, 2, 4, 0, 2, -1, -1, -1, -1, -1, -1, -1},
{2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8, -1, -1, -1, -1},
{9, 5, 4, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 11, 2, 0, 8, 11, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1},
{0, 5, 4, 0, 1, 5, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1},
{2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5, -1, -1, -1, -1},
{10, 3, 11, 10, 1, 3, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1},
{4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10, -1, -1, -1, -1},
{5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3, -1, -1, -1, -1},
{5, 4, 8, 5, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1},
{9, 7, 8, 5, 7, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{9, 3, 0, 9, 5, 3, 5, 7, 3, -1, -1, -1, -1, -1, -1, -1},
{0, 7, 8, 0, 1, 7, 1, 5, 7, -1, -1, -1, -1, -1, -1, -1},
{1, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{9, 7, 8, 9, 5, 7, 10, 1, 2, -1, -1, -1, -1, -1, -1, -1},
{10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3, -1, -1, -1, -1},
{8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2, -1, -1, -1, -1},
{2, 10, 5, 2, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1},
{7, 9, 5, 7, 8, 9, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1},
{9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11, -1, -1, -1, -1},
{2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7, -1, -1, -1, -1},
{11, 2, 1, 11, 1, 7, 7, 1, 5, -1, -1, -1, -1, -1, -1, -1},
{9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11, -1, -1, -1, -1},
{5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0, -1},
{11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0, -1},
{11, 10, 5, 7, 11, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 8, 3, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{9, 0, 1, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 8, 3, 1, 9, 8, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1},
{1, 6, 5, 2, 6, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 6, 5, 1, 2, 6, 3, 0, 8, -1, -1, -1, -1, -1, -1, -1},
{9, 6, 5, 9, 0, 6, 0, 2, 6, -1, -1, -1, -1, -1, -1, -1},
{5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8, -1, -1, -1, -1},
{2, 3, 11, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{11, 0, 8, 11, 2, 0, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1},
{0, 1, 9, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1},
{5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11, -1, -1, -1, -1},
{6, 3, 11, 6, 5, 3, 5, 1, 3, -1, -1, -1, -1, -1, -1, -1},
{0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6, -1, -1, -1, -1},
{3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9, -1, -1, -1, -1},
{6, 5, 9, 6, 9, 11, 11, 9, 8, -1, -1, -1, -1, -1, -1, -1},
{5, 10, 6, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{4, 3, 0, 4, 7, 3, 6, 5, 10, -1, -1, -1, -1, -1, -1, -1},
{1, 9, 0, 5, 10, 6, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1},
{10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4, -1, -1, -1, -1},
{6, 1, 2, 6, 5, 1, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1},
{1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7, -1, -1, -1, -1},
{8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6, -1, -1, -1, -1},
{7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9, -1},
{3, 11, 2, 7, 8, 4, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1},
{5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11, -1, -1, -1, -1},
{0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1},
{9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6, -1},
{8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6, -1, -1, -1, -1},
{5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11, -1},
{0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7, -1},
{6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9, -1, -1, -1, -1},
{10, 4, 9, 6, 4, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{4, 10, 6, 4, 9, 10, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1},
{10, 0, 1, 10, 6, 0, 6, 4, 0, -1, -1, -1, -1, -1, -1, -1},
{8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10, -1, -1, -1, -1},
{1, 4, 9, 1, 2, 4, 2, 6, 4, -1, -1, -1, -1, -1, -1, -1},
{3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4, -1, -1, -1, -1},
{0, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{8, 3, 2, 8, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1},
{10, 4, 9, 10, 6, 4, 11, 2, 3, -1, -1, -1, -1, -1, -1, -1},
{0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6, -1, -1, -1, -1},
{3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10, -1, -1, -1, -1},
{6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1, -1},
{9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3, -1, -1, -1, -1},
{8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1, -1},
{3, 11, 6, 3, 6, 0, 0, 6, 4, -1, -1, -1, -1, -1, -1, -1},
{6, 4, 8, 11, 6, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{7, 10, 6, 7, 8, 10, 8, 9, 10, -1, -1, -1, -1, -1, -1, -1},
{0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10, -1, -1, -1, -1},
{10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0, -1, -1, -1, -1},
{10, 6, 7, 10, 7, 1, 1, 7, 3, -1, -1, -1, -1, -1, -1, -1},
{1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7, -1, -1, -1, -1},
{2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9, -1},
{7, 8, 0, 7, 0, 6, 6, 0, 2, -1, -1, -1, -1, -1, -1, -1},
{7, 3, 2, 6, 7, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7, -1, -1, -1, -1},
{2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7, -1},
{1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11, -1},
{11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1, -1, -1, -1, -1},
{8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6, -1},
{0, 9, 1, 11, 6, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0, -1, -1, -1, -1},
{7, 11, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{3, 0, 8, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 1, 9, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{8, 1, 9, 8, 3, 1, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1},
{10, 1, 2, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 2, 10, 3, 0, 8, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1},
{2, 9, 0, 2, 10, 9, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1},
{6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8, -1, -1, -1, -1},
{7, 2, 3, 6, 2, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{7, 0, 8, 7, 6, 0, 6, 2, 0, -1, -1, -1, -1, -1, -1, -1},
{2, 7, 6, 2, 3, 7, 0, 1, 9, -1, -1, -1, -1, -1, -1, -1},
{1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6, -1, -1, -1, -1},
{10, 7, 6, 10, 1, 7, 1, 3, 7, -1, -1, -1, -1, -1, -1, -1},
{10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8, -1, -1, -1, -1},
{0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7, -1, -1, -1, -1},
{7, 6, 10, 7, 10, 8, 8, 10, 9, -1, -1, -1, -1, -1, -1, -1},
{6, 8, 4, 11, 8, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{3, 6, 11, 3, 0, 6, 0, 4, 6, -1, -1, -1, -1, -1, -1, -1},
{8, 6, 11, 8, 4, 6, 9, 0, 1, -1, -1, -1, -1, -1, -1, -1},
{9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6, -1, -1, -1, -1},
{6, 8, 4, 6, 11, 8, 2, 10, 1, -1, -1, -1, -1, -1, -1, -1},
{1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6, -1, -1, -1, -1},
{4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9, -1, -1, -1, -1},
{10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3, -1},
{8, 2, 3, 8, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1},
{0, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8, -1, -1, -1, -1},
{1, 9, 4, 1, 4, 2, 2, 4, 6, -1, -1, -1, -1, -1, -1, -1},
{8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1, -1, -1, -1, -1},
{10, 1, 0, 10, 0, 6, 6, 0, 4, -1, -1, -1, -1, -1, -1, -1},
{4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3, -1},
{10, 9, 4, 6, 10, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{4, 9, 5, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 8, 3, 4, 9, 5, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1},
{5, 0, 1, 5, 4, 0, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1},
{11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5, -1, -1, -1, -1},
{9, 5, 4, 10, 1, 2, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1},
{6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5, -1, -1, -1, -1},
{7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2, -1, -1, -1, -1},
{3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6, -1},
{7, 2, 3, 7, 6, 2, 5, 4, 9, -1, -1, -1, -1, -1, -1, -1},
{9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7, -1, -1, -1, -1},
{3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0, -1, -1, -1, -1},
{6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8, -1},
{9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7, -1, -1, -1, -1},
{1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4, -1},
{4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10, -1},
{7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10, -1, -1, -1, -1},
{6, 9, 5, 6, 11, 9, 11, 8, 9, -1, -1, -1, -1, -1, -1, -1},
{3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5, -1, -1, -1, -1},
{0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11, -1, -1, -1, -1},
{6, 11, 3, 6, 3, 5, 5, 3, 1, -1, -1, -1, -1, -1, -1, -1},
{1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6, -1, -1, -1, -1},
{0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10, -1},
{11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5, -1},
{6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3, -1, -1, -1, -1},
{5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2, -1, -1, -1, -1},
{9, 5, 6, 9, 6, 0, 0, 6, 2, -1, -1, -1, -1, -1, -1, -1},
{1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8, -1},
{1, 5, 6, 2, 1, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6, -1},
{10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0, -1, -1, -1, -1},
{0, 3, 8, 5, 6, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{10, 5, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{11, 5, 10, 7, 5, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{11, 5, 10, 11, 7, 5, 8, 3, 0, -1, -1, -1, -1, -1, -1, -1},
{5, 11, 7, 5, 10, 11, 1, 9, 0, -1, -1, -1, -1, -1, -1, -1},
{10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1, -1, -1, -1, -1},
{11, 1, 2, 11, 7, 1, 7, 5, 1, -1, -1, -1, -1, -1, -1, -1},
{0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11, -1, -1, -1, -1},
{9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7, -1, -1, -1, -1},
{7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2, -1},
{2, 5, 10, 2, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1},
{8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5, -1, -1, -1, -1},
{9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2, -1, -1, -1, -1},
{9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2, -1},
{1, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 8, 7, 0, 7, 1, 1, 7, 5, -1, -1, -1, -1, -1, -1, -1},
{9, 0, 3, 9, 3, 5, 5, 3, 7, -1, -1, -1, -1, -1, -1, -1},
{9, 8, 7, 5, 9, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{5, 8, 4, 5, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1},
{5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0, -1, -1, -1, -1},
{0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5, -1, -1, -1, -1},
{10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4, -1},
{2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8, -1, -1, -1, -1},
{0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11, -1},
{0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5, -1},
{9, 4, 5, 2, 11, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4, -1, -1, -1, -1},
{5, 10, 2, 5, 2, 4, 4, 2, 0, -1, -1, -1, -1, -1, -1, -1},
{3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9, -1},
{5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2, -1, -1, -1, -1},
{8, 4, 5, 8, 5, 3, 3, 5, 1, -1, -1, -1, -1, -1, -1, -1},
{0, 4, 5, 1, 0, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5, -1, -1, -1, -1},
{9, 4, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{4, 11, 7, 4, 9, 11, 9, 10, 11, -1, -1, -1, -1, -1, -1, -1},
{0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11, -1, -1, -1, -1},
{1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11, -1, -1, -1, -1},
{3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4, -1},
{4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2, -1, -1, -1, -1},
{9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3, -1},
{11, 7, 4, 11, 4, 2, 2, 4, 0, -1, -1, -1, -1, -1, -1, -1},
{11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4, -1, -1, -1, -1},
{2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9, -1, -1, -1, -1},
{9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7, -1},
{3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10, -1},
{1, 10, 2, 8, 7, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{4, 9, 1, 4, 1, 7, 7, 1, 3, -1, -1, -1, -1, -1, -1, -1},
{4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1, -1, -1, -1, -1},
{4, 0, 3, 7, 4, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{4, 8, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{9, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{3, 0, 9, 3, 9, 11, 11, 9, 10, -1, -1, -1, -1, -1, -1, -1},
{0, 1, 10, 0, 10, 8, 8, 10, 11, -1, -1, -1, -1, -1, -1, -1},
{3, 1, 10, 11, 3, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 2, 11, 1, 11, 9, 9, 11, 8, -1, -1, -1, -1, -1, -1, -1},
{3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9, -1, -1, -1, -1},
{0, 2, 11, 8, 0, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{3, 2, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{2, 3, 8, 2, 8, 10, 10, 8, 9, -1, -1, -1, -1, -1, -1, -1},
{9, 10, 2, 0, 9, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8, -1, -1, -1, -1},
{1, 10, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{1, 3, 8, 9, 1, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 9, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{0, 3, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
{-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}};

 有了这张表格之后，我们就可以渲染任意一个等值面了。比如下面的这个游泳圈和爱心。

 (x^2+9/4y^2+z^2-1)^3=(x^2z^2+9y^2z^3)/80

 完整代码如下:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class NewBehaviourScript : MonoBehaviour
{
    // Start is called before the first frame update
    List<Vector3> points=new List<Vector3>();
    List<int> tri=new List<int>();
    List<int>[] point2rti;
    int point_index=0;
    float size = 0.05f;
    bool Isin(Vector3 pos)
    {
       
        if (value_in_ring(pos) > 0)
            return false;
        else return true;

    }
    float value_in_ring(Vector3 pos)
    {
        float R = 1f;
        float r = 0.5f;
        float res = Mathf.Pow((R - Mathf.Pow((pos.x * pos.x + pos.y * pos.y), 0.5f)), 2) + pos.z * pos.z - r * r;
        return res;
    }
    float value_in_heart(Vector3 pos)
    {
        float R = 1f;
        float r = 0.5f;
        float x = pos.x;
        float y = pos.y;
        float z = pos.z;
        float res = Mathf.Pow((x*x+9/4*y*y+z*z-1), 3) -x*x*z*z*z+9/80*y*y*z*z*z;
        return res;
    }
    class Cube
    {
        float size;
        public static byte VULF = 1 << 0;
        public static byte VULB = 1 << 1;
        public static byte VLLB = 1 << 2;
        public static byte VLLF = 1 << 3;
        public static byte VURF = 1 << 4;
        public static byte VURB = 1 << 5;
        public static byte VLRB = 1 << 6;
        public static byte VLRF = 1 << 7;
        //以上为体素为实点的位标记
        public static Vector3Int[] PointIndexToPointDelta = new Vector3Int[8]
        {
        new Vector3Int(0, 1, 1 ),
        new Vector3Int(0, 1, 0 ),
        new Vector3Int(0, 0, 0 ),
        new Vector3Int(0, 0, 1 ),
        new Vector3Int(1, 1, 1 ),
        new Vector3Int(1, 1, 0 ),
        new Vector3Int(1, 0, 0 ),
        new Vector3Int(1, 0, 1 )
        };//体元内每个体素相对基准体素坐标的偏移
        public static byte[] PointIndexToFlag = new byte[8]
        {
        VULF,
        VULB,
        VLLB,
        VLLF,
        VURF,
        VURB,
        VLRB,
        VLRF
        };//每个体素对应的位标记
        public static int[,] EdgeIndexToEdgeVertexIndex = new int[12, 2]
        {
        {0,1}, {1,2},
        {2,3},{3,0},
        {4,5},{5,6},
        {6,7}, {7,4},
        {0,4}, {1,5},
        {2,6}, {3,7}
        };//每个边对应的两顶点体素的索引
        public int CellIndexX;
        public int CellIndexY;
        public int CellIndexZ;
        public Cube(int cellIndexX, int cellIndexY, int cellIndexZ,float size)
        {

            this.size = size;
            this.CellIndexX = cellIndexX;
            this.CellIndexY = cellIndexY;
            this.CellIndexZ = cellIndexZ;
            for (int i = 0; i < 8; i++)
            {
                cubeImageIndices[i].x = (cellIndexX + PointIndexToPointDelta[i].x)*size;
                cubeImageIndices[i].y = (cellIndexY + PointIndexToPointDelta[i].y)* size;
                cubeImageIndices[i].z = (cellIndexZ + PointIndexToPointDelta[i].z)* size;
            }
        }//使用基准体素坐标初始化Cube
        public Vector3[] cubeImageIndices = new Vector3[8];


    };
    private byte GetConfig(ref Cube cube)
    {
        byte value = 0;
        for (int i = 0; i < 8; i++)
        {
            if (Isin(new Vector3( cube.cubeImageIndices[i].x, cube.cubeImageIndices[i].y, cube.cubeImageIndices[i].z)))
            {
                value |= Cube.PointIndexToFlag[i];
            }
        }
        return value;
    }//get copnfig

    Vector3 interpolation(Vector3 p1,Vector3 p2)
    {
        
        float v1 = value_in_ring(p1);
        float v2 = value_in_ring(p2);
        Vector3 P = p1 + (0 - v1) * (p2 - p1) / (v2 - v1);

        return P;
    }

    void ExtractTriangles(ref Cube cube, byte value)
    {



        if (TriTable[value, 0] != -1)
        {
            int index = 0;
            while (TriTable[value, index] != -1)
            {
                int e0index = TriTable[value, index];
                int e1index = TriTable[value, index + 1];
                int e2index = TriTable[value, index + 2];

                Vector3 e0p0 = cube.cubeImageIndices[Cube.EdgeIndexToEdgeVertexIndex[e0index, 0]];//静态变量应该用类来访问，而不是实例化变量来方位
                Vector3 e0p1 = cube.cubeImageIndices[Cube.EdgeIndexToEdgeVertexIndex[e0index, 1]];

                Vector3 e1p0 = cube.cubeImageIndices[Cube.EdgeIndexToEdgeVertexIndex[e1index, 0]];
                Vector3 e1p1 = cube.cubeImageIndices[Cube.EdgeIndexToEdgeVertexIndex[e1index, 1]];

                Vector3 e2p0 = cube.cubeImageIndices[Cube.EdgeIndexToEdgeVertexIndex[e2index, 0]];
                Vector3 e2p1 = cube.cubeImageIndices[Cube.EdgeIndexToEdgeVertexIndex[e2index, 1]];



                Vector3 e0pm = interpolation(e0p0, e0p1);
                Vector3 e1pm = interpolation(e1p0 , e1p1);
                Vector3 e2pm = interpolation(e2p0 ,e2p1);
               
                points.Add(e0pm);
                points.Add(e1pm);
                points.Add(e2pm);
                tri.Add(point_index);
                tri.Add(point_index + 1);
                tri.Add(point_index + 2);
                point_index += 3;
                index += 3;
            }
        }


     







    }
    void Start()
    {
        int w=60;
        int h=60;
        int d =60;
        
        for (int i = w/2-w; i < w/2; i++)
        {
            for(int j=h/2-h;j<h/2;j++)
            {

                for(int k=d/2-d;k<d/2;k++)
                {
                    Cube acube = new Cube(i,j,k, size);//生成一个立方体
                    byte config = GetConfig(ref acube);//进行立方体的配置
                    ExtractTriangles(ref acube, config);
                }
            }
        }
        point2rti = new List<int>[points.Count];
        //遍历三角面片，计算point2tri;
        for (int i = 0; i < tri.Count; i = i + 3)
        {
            //i是面片索引，tri(i),tri(i+1),tri(i+2)是改面片的点的索引
            //对于每个面将其索引添加到点对应的list里
            point2rti[tri[i]] = new List<int>();
            point2rti[tri[i+1]] = new List<int>();
            point2rti[tri[i+2]] = new List<int>();
            point2rti[tri[i]].Add(i);
            point2rti[tri[i + 1]].Add(i);
            point2rti[tri[i + 2]].Add(i);
        }


        //拉普拉斯平滑处理
        int time = 500;
        for (int i = 0; i < time; i++)
        {
            L_smooth(ref points, ref tri, ref point2rti);
        }
      


        gameObject.AddComponent<MeshFilter>();
        gameObject.AddComponent<MeshRenderer>();
        Mesh mesh = gameObject.GetComponent<MeshFilter>().mesh;
        mesh.indexFormat = UnityEngine.Rendering.IndexFormat.UInt32;
        mesh.vertices = points.ToArray();
        mesh.triangles = tri.ToArray();
        mesh.RecalculateNormals();//法线渲染





    }

    // Update is called once per frame
    void Update()
    {
        
    }

    public static int[,] TriTable = new int[256, 16]
        {
                {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 8, 3, 9, 8, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 8, 3, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {9, 2, 10, 0, 2, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {2, 8, 3, 2, 10, 8, 10, 9, 8, -1, -1, -1, -1, -1, -1, -1},
                {3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 11, 2, 8, 11, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 9, 0, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 11, 2, 1, 9, 11, 9, 8, 11, -1, -1, -1, -1, -1, -1, -1},
                {3, 10, 1, 11, 10, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 10, 1, 0, 8, 10, 8, 11, 10, -1, -1, -1, -1, -1, -1, -1},
                {3, 9, 0, 3, 11, 9, 11, 10, 9, -1, -1, -1, -1, -1, -1, -1},
                {9, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {4, 3, 0, 7, 3, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 1, 9, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {4, 1, 9, 4, 7, 1, 7, 3, 1, -1, -1, -1, -1, -1, -1, -1},
                {1, 2, 10, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {3, 4, 7, 3, 0, 4, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1},
                {9, 2, 10, 9, 0, 2, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1},
                {2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4, -1, -1, -1, -1},
                {8, 4, 7, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {11, 4, 7, 11, 2, 4, 2, 0, 4, -1, -1, -1, -1, -1, -1, -1},
                {9, 0, 1, 8, 4, 7, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1},
                {4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1, -1, -1, -1, -1},
                {3, 10, 1, 3, 11, 10, 7, 8, 4, -1, -1, -1, -1, -1, -1, -1},
                {1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4, -1, -1, -1, -1},
                {4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3, -1, -1, -1, -1},
                {4, 7, 11, 4, 11, 9, 9, 11, 10, -1, -1, -1, -1, -1, -1, -1},
                {9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {9, 5, 4, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 5, 4, 1, 5, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {8, 5, 4, 8, 3, 5, 3, 1, 5, -1, -1, -1, -1, -1, -1, -1},
                {1, 2, 10, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {3, 0, 8, 1, 2, 10, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1},
                {5, 2, 10, 5, 4, 2, 4, 0, 2, -1, -1, -1, -1, -1, -1, -1},
                {2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8, -1, -1, -1, -1},
                {9, 5, 4, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 11, 2, 0, 8, 11, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1},
                {0, 5, 4, 0, 1, 5, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1},
                {2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5, -1, -1, -1, -1},
                {10, 3, 11, 10, 1, 3, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1},
                {4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10, -1, -1, -1, -1},
                {5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3, -1, -1, -1, -1},
                {5, 4, 8, 5, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1},
                {9, 7, 8, 5, 7, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {9, 3, 0, 9, 5, 3, 5, 7, 3, -1, -1, -1, -1, -1, -1, -1},
                {0, 7, 8, 0, 1, 7, 1, 5, 7, -1, -1, -1, -1, -1, -1, -1},
                {1, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {9, 7, 8, 9, 5, 7, 10, 1, 2, -1, -1, -1, -1, -1, -1, -1},
                {10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3, -1, -1, -1, -1},
                {8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2, -1, -1, -1, -1},
                {2, 10, 5, 2, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1},
                {7, 9, 5, 7, 8, 9, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1},
                {9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11, -1, -1, -1, -1},
                {2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7, -1, -1, -1, -1},
                {11, 2, 1, 11, 1, 7, 7, 1, 5, -1, -1, -1, -1, -1, -1, -1},
                {9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11, -1, -1, -1, -1},
                {5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0, -1},
                {11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0, -1},
                {11, 10, 5, 7, 11, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 8, 3, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {9, 0, 1, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 8, 3, 1, 9, 8, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1},
                {1, 6, 5, 2, 6, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 6, 5, 1, 2, 6, 3, 0, 8, -1, -1, -1, -1, -1, -1, -1},
                {9, 6, 5, 9, 0, 6, 0, 2, 6, -1, -1, -1, -1, -1, -1, -1},
                {5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8, -1, -1, -1, -1},
                {2, 3, 11, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {11, 0, 8, 11, 2, 0, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1},
                {0, 1, 9, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1},
                {5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11, -1, -1, -1, -1},
                {6, 3, 11, 6, 5, 3, 5, 1, 3, -1, -1, -1, -1, -1, -1, -1},
                {0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6, -1, -1, -1, -1},
                {3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9, -1, -1, -1, -1},
                {6, 5, 9, 6, 9, 11, 11, 9, 8, -1, -1, -1, -1, -1, -1, -1},
                {5, 10, 6, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {4, 3, 0, 4, 7, 3, 6, 5, 10, -1, -1, -1, -1, -1, -1, -1},
                {1, 9, 0, 5, 10, 6, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1},
                {10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4, -1, -1, -1, -1},
                {6, 1, 2, 6, 5, 1, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1},
                {1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7, -1, -1, -1, -1},
                {8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6, -1, -1, -1, -1},
                {7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9, -1},
                {3, 11, 2, 7, 8, 4, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1},
                {5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11, -1, -1, -1, -1},
                {0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1},
                {9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6, -1},
                {8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6, -1, -1, -1, -1},
                {5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11, -1},
                {0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7, -1},
                {6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9, -1, -1, -1, -1},
                {10, 4, 9, 6, 4, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {4, 10, 6, 4, 9, 10, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1},
                {10, 0, 1, 10, 6, 0, 6, 4, 0, -1, -1, -1, -1, -1, -1, -1},
                {8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10, -1, -1, -1, -1},
                {1, 4, 9, 1, 2, 4, 2, 6, 4, -1, -1, -1, -1, -1, -1, -1},
                {3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4, -1, -1, -1, -1},
                {0, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {8, 3, 2, 8, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1},
                {10, 4, 9, 10, 6, 4, 11, 2, 3, -1, -1, -1, -1, -1, -1, -1},
                {0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6, -1, -1, -1, -1},
                {3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10, -1, -1, -1, -1},
                {6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1, -1},
                {9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3, -1, -1, -1, -1},
                {8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1, -1},
                {3, 11, 6, 3, 6, 0, 0, 6, 4, -1, -1, -1, -1, -1, -1, -1},
                {6, 4, 8, 11, 6, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {7, 10, 6, 7, 8, 10, 8, 9, 10, -1, -1, -1, -1, -1, -1, -1},
                {0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10, -1, -1, -1, -1},
                {10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0, -1, -1, -1, -1},
                {10, 6, 7, 10, 7, 1, 1, 7, 3, -1, -1, -1, -1, -1, -1, -1},
                {1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7, -1, -1, -1, -1},
                {2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9, -1},
                {7, 8, 0, 7, 0, 6, 6, 0, 2, -1, -1, -1, -1, -1, -1, -1},
                {7, 3, 2, 6, 7, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7, -1, -1, -1, -1},
                {2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7, -1},
                {1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11, -1},
                {11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1, -1, -1, -1, -1},
                {8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6, -1},
                {0, 9, 1, 11, 6, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0, -1, -1, -1, -1},
                {7, 11, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {3, 0, 8, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 1, 9, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {8, 1, 9, 8, 3, 1, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1},
                {10, 1, 2, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 2, 10, 3, 0, 8, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1},
                {2, 9, 0, 2, 10, 9, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1},
                {6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8, -1, -1, -1, -1},
                {7, 2, 3, 6, 2, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {7, 0, 8, 7, 6, 0, 6, 2, 0, -1, -1, -1, -1, -1, -1, -1},
                {2, 7, 6, 2, 3, 7, 0, 1, 9, -1, -1, -1, -1, -1, -1, -1},
                {1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6, -1, -1, -1, -1},
                {10, 7, 6, 10, 1, 7, 1, 3, 7, -1, -1, -1, -1, -1, -1, -1},
                {10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8, -1, -1, -1, -1},
                {0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7, -1, -1, -1, -1},
                {7, 6, 10, 7, 10, 8, 8, 10, 9, -1, -1, -1, -1, -1, -1, -1},
                {6, 8, 4, 11, 8, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {3, 6, 11, 3, 0, 6, 0, 4, 6, -1, -1, -1, -1, -1, -1, -1},
                {8, 6, 11, 8, 4, 6, 9, 0, 1, -1, -1, -1, -1, -1, -1, -1},
                {9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6, -1, -1, -1, -1},
                {6, 8, 4, 6, 11, 8, 2, 10, 1, -1, -1, -1, -1, -1, -1, -1},
                {1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6, -1, -1, -1, -1},
                {4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9, -1, -1, -1, -1},
                {10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3, -1},
                {8, 2, 3, 8, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1},
                {0, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8, -1, -1, -1, -1},
                {1, 9, 4, 1, 4, 2, 2, 4, 6, -1, -1, -1, -1, -1, -1, -1},
                {8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1, -1, -1, -1, -1},
                {10, 1, 0, 10, 0, 6, 6, 0, 4, -1, -1, -1, -1, -1, -1, -1},
                {4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3, -1},
                {10, 9, 4, 6, 10, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {4, 9, 5, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 8, 3, 4, 9, 5, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1},
                {5, 0, 1, 5, 4, 0, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1},
                {11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5, -1, -1, -1, -1},
                {9, 5, 4, 10, 1, 2, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1},
                {6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5, -1, -1, -1, -1},
                {7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2, -1, -1, -1, -1},
                {3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6, -1},
                {7, 2, 3, 7, 6, 2, 5, 4, 9, -1, -1, -1, -1, -1, -1, -1},
                {9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7, -1, -1, -1, -1},
                {3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0, -1, -1, -1, -1},
                {6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8, -1},
                {9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7, -1, -1, -1, -1},
                {1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4, -1},
                {4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10, -1},
                {7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10, -1, -1, -1, -1},
                {6, 9, 5, 6, 11, 9, 11, 8, 9, -1, -1, -1, -1, -1, -1, -1},
                {3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5, -1, -1, -1, -1},
                {0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11, -1, -1, -1, -1},
                {6, 11, 3, 6, 3, 5, 5, 3, 1, -1, -1, -1, -1, -1, -1, -1},
                {1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6, -1, -1, -1, -1},
                {0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10, -1},
                {11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5, -1},
                {6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3, -1, -1, -1, -1},
                {5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2, -1, -1, -1, -1},
                {9, 5, 6, 9, 6, 0, 0, 6, 2, -1, -1, -1, -1, -1, -1, -1},
                {1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8, -1},
                {1, 5, 6, 2, 1, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6, -1},
                {10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0, -1, -1, -1, -1},
                {0, 3, 8, 5, 6, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {10, 5, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {11, 5, 10, 7, 5, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {11, 5, 10, 11, 7, 5, 8, 3, 0, -1, -1, -1, -1, -1, -1, -1},
                {5, 11, 7, 5, 10, 11, 1, 9, 0, -1, -1, -1, -1, -1, -1, -1},
                {10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1, -1, -1, -1, -1},
                {11, 1, 2, 11, 7, 1, 7, 5, 1, -1, -1, -1, -1, -1, -1, -1},
                {0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11, -1, -1, -1, -1},
                {9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7, -1, -1, -1, -1},
                {7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2, -1},
                {2, 5, 10, 2, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1},
                {8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5, -1, -1, -1, -1},
                {9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2, -1, -1, -1, -1},
                {9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2, -1},
                {1, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 8, 7, 0, 7, 1, 1, 7, 5, -1, -1, -1, -1, -1, -1, -1},
                {9, 0, 3, 9, 3, 5, 5, 3, 7, -1, -1, -1, -1, -1, -1, -1},
                {9, 8, 7, 5, 9, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {5, 8, 4, 5, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1},
                {5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0, -1, -1, -1, -1},
                {0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5, -1, -1, -1, -1},
                {10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4, -1},
                {2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8, -1, -1, -1, -1},
                {0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11, -1},
                {0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5, -1},
                {9, 4, 5, 2, 11, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4, -1, -1, -1, -1},
                {5, 10, 2, 5, 2, 4, 4, 2, 0, -1, -1, -1, -1, -1, -1, -1},
                {3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9, -1},
                {5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2, -1, -1, -1, -1},
                {8, 4, 5, 8, 5, 3, 3, 5, 1, -1, -1, -1, -1, -1, -1, -1},
                {0, 4, 5, 1, 0, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5, -1, -1, -1, -1},
                {9, 4, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {4, 11, 7, 4, 9, 11, 9, 10, 11, -1, -1, -1, -1, -1, -1, -1},
                {0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11, -1, -1, -1, -1},
                {1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11, -1, -1, -1, -1},
                {3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4, -1},
                {4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2, -1, -1, -1, -1},
                {9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3, -1},
                {11, 7, 4, 11, 4, 2, 2, 4, 0, -1, -1, -1, -1, -1, -1, -1},
                {11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4, -1, -1, -1, -1},
                {2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9, -1, -1, -1, -1},
                {9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7, -1},
                {3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10, -1},
                {1, 10, 2, 8, 7, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {4, 9, 1, 4, 1, 7, 7, 1, 3, -1, -1, -1, -1, -1, -1, -1},
                {4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1, -1, -1, -1, -1},
                {4, 0, 3, 7, 4, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {4, 8, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {9, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {3, 0, 9, 3, 9, 11, 11, 9, 10, -1, -1, -1, -1, -1, -1, -1},
                {0, 1, 10, 0, 10, 8, 8, 10, 11, -1, -1, -1, -1, -1, -1, -1},
                {3, 1, 10, 11, 3, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 2, 11, 1, 11, 9, 9, 11, 8, -1, -1, -1, -1, -1, -1, -1},
                {3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9, -1, -1, -1, -1},
                {0, 2, 11, 8, 0, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {3, 2, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {2, 3, 8, 2, 8, 10, 10, 8, 9, -1, -1, -1, -1, -1, -1, -1},
                {9, 10, 2, 0, 9, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8, -1, -1, -1, -1},
                {1, 10, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {1, 3, 8, 9, 1, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 9, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {0, 3, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1},
                {-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1}
        };
    void mesh_optimization(ref List<Vector3> points ,ref List<int> tri, ref List<int> point2rti)//输入参数为点数据与网格数据
    {
      
        
        //消减面片
        float min_area = 0.1f * (0.25f * size * size);
        for (int i = 0; i < tri.Count; i = i + 3)
        {

            Vector3 p1 = points[tri[i]];
            Vector3 p2 = points[tri[i+1]];
            Vector3 p3 = points[tri[i+1]];

            float area = calculate_area(p1, p2, p3);
            if (area < min_area)//如果三角型的面积小于最小面积，消除这个三角形
            {

                Vector3 p = (p1 + p2 + p3) / 3;
            }


        }


    }
    void L_smooth(ref List<Vector3> points, ref List<int> tri, ref List<int>[] point2rti)
    {
        for (int i = 0; i < points.Count; i++)
        {

            Vector3 temp = new Vector3(0, 0, 0);
            int count = 0;
            for (int j = 0; j < point2rti[i].Count; j++)// point2rti[i][j]为包含i节点的第j个三角形
            {

          
                for(int k=0;k<3;k++)
                {
                    int ii= tri[point2rti[i][j]+k];
                    if (ii != i)
                    {

                        temp +=0.00005f*( points[ii]-points[i]);
                        count++;

                    }


                }

            }

            points[i] = points[i]+ temp / (float)count;
        }

    }

    float calculate_area(Vector3 p1,Vector3 p2,Vector3 p3)
    {
        
        float a = (p1 - p2).magnitude;
        float b = (p2 - p3).magnitude;
        float c = (p1 - p3).magnitude;
        float p = (a + b + c) * 0.5f;
        float area = Mathf.Pow(p*(p-a)*(p-b)*(p-c),0.5f);
        return area;

    }
}

本文内容由网友自发贡献，版权归原作者所有，本站不承担相应法律责任。如您发现有涉嫌抄袭侵权的内容，请联系:hwhale#tublm.com(使用前将#替换为@)

几何学

图形学

人类高质量网格重建算法 的相关文章

随机推荐

  • 自己动手写CSDN博客提取器源码分析之三：处理网页保存为pdf文件

    转载自 http blog csdn net w397090770 article details 7768129 下面我讲下处理pdf文件的 这里我用了PD4ML来处理的 原因有几个 1 它对CSS的支持做的很好 2 可以处理图片 很爽吧

  • python——获取当前文件路径

    1 os getcwd os getcwd 不需要参数 os getcwd 获取的是当前工作的路径 也就是脚本运行的地方 而不是脚本本身所存在的地方 代码 print os getcwd 测试结果 2 os path abspath os

  • npm ERR! code ELIFECYCLE npm ERR! errno 1 npm ERR! webpack_test@1.0.0 dev: `webpack` npm ERR! Exit s

    webpack打包时 报如图错误 问题产生的原因 webpack版本过高 导致npm i 时安装的依赖不再出现在node modules中 从而命令执行报错 解决方案1 运行npm i dev 安装一下开发环境下的依赖即可 随后再执行需要运

  • 云计算虚拟机技术-KVM安装

    云计算虚拟机技术 KVM安装 作者 尹正杰 版权声明 原创作品 谢绝转载 否则将追究法律责任 身为运维的小伙伴估计大家都清楚KVM 因为在CentOS里面KVM还算很折腾的一个软件 早期CentOS使用的xen虚拟化 后因收购KVM 于是就

  • 程序员工具网站（比较有意思）

    http www cxy521 com index html

  • html中中文显示乱码,网页中文显示乱码问题

    大家好 本人目前在学习ASP NET MVC5 出现了网页显示中文乱码的问题 诡异的是Home Index页面下的中文是可以正常显示的 其他页面的中文就不能正常显示了 不能正常显示中文的页面中 通过后台代码显示的字符串是正常的 直接卸载HT

  • Oracle数据库 Explain工具（语句执行索引使用情况）

    1 Explain工具的作用 Explain工具可以查看SQL语句的执行过程SQL索引的使用情况 2 Oracle数据库查看语句执行的情况语句 explain plan for select id from abc where a 1 an

  • 不确定性世界中的行为预测

    桔妹导读 自动驾驶系统中的行为预测模块主要负责预判其他交通参与者未来一段时间的行为 其困难主要源自现实世界和人类行为的高度不确定性 本文列举了现实环境中的不同场景 讨论了两种不同预测方法 即基于行为规划的预测和基于机器学习的预测 用于准确预

  • 多元统计分析实验-线性回归

    1 实验目的 通过本实验使学生熟练运用python语言编写线性回归分析程序 2 实验内容 有以下两组数据 第一组数据为我国2004年31个地区城镇居民年人均可支配收入和年人均消费性支出数据 第二组数据为随机抽查的36个人的当前年薪y 美元

  • 基于直方图均衡化的水下激光图像处理（MATLAB仿真）

    原文 http blog csdn net sunlinyi66 article details 39215447 基于直方图均衡化的水下图像增强 MATLAB仿真 可以对RGB三个通道进行直方图规定化从而实现水下图像增强 此处我们使用的是

  • python爬虫学习34

    python爬虫学习34 目录 python爬虫学习34 九 parsel 库的使用其一 9 1 parsel 库的下载 9 2 初始化 9 3 提取文本 今天呢就进入到爬虫网页数据解析提取的最后一个小节 与大家一同学习parsel库的使用

  • 牛客网 D-坐标计数

    链接 https ac nowcoder com acm contest 10746 D 来源 牛客网 题目描述 定义一个坐标变换 坐标 x y 变换后变为 x y x y 给定一片矩形区域 计算区域内有多少个整数点在经过有限次变换后变为

  • redis持久化机制

    redis持久化机制 https www jianshu com p 9d6cebe98d5a

  • C# - var 关键字

    C var 关键字 引言 var 关键字使用规则 var 在表达式中的应用 例 1 var 可以选 例 2 var 必要 引言 C 代码经常会看到 var 关键字定义一个变量 带点神秘色彩 今天就来揭秘一下 从 C 3 0 开始 在方法范围

  • 【2022年1月1日更新】宝塔+青龙+Nvjdc&一对一推送+wskey智能化+管理员推送+Ninja+SaoBing面板+kingfeng+go-cqhttp/qqbot

    前言 2023年这些技术一部分不能用了 现在最优的技术是 青龙 ark短信登录 傻妞机器人 cqhttp或one node 具体可以参考我专栏里的其他文章 专栏 青龙相关 首先声明我也不是什么大佬 只是坑踩多了有经验了而已 看了无数大佬的文

  • GitHub上排名前100的优秀开源项目

    主要对当前 GitHub 排名前 100 的项目做一个简单的简介 方便初学者快速了解到当前 Objective C 在 GitHub 的情况 若有任何疑问可通过微博 甲骨文儿联系我 项目名称 项目信息 1 AFNetworking 作者是

  • 文件系统（4）——Ubuntu system service设置自动登录

    参考链接 linux systemd启动方式解析 如何开机自动登录 开机自动运行程序 https blog csdn net a617996505 article details 88423794 涉及文件 OK3399 linux ubu

  • STM32学习笔记----（4）GPIO

    STM32必须掌握的有时钟树和系统架构 感觉那个是比较简单的 多看几遍就能看会了 在后面的笔记中也会记录到这些 如果自己看不懂的话可以跟着不同外设的笔记来了解一下 这篇笔记是讲解的GPIO 学会了GPIO的话就可以完成32的第一个功能 ST

  • Android应用的闪退（crash）分析

    文章转自阿里客户端工程师试题简析 Android应用的闪退 crash 分析 1 问题描述 闪退 Crash 是客户端程序在运行时遭遇无法处理的异常或错误时而退出应用程序的表现 请从crash发生的原因分类与解决方法 在出现crash后如何

  • 人类高质量网格重建算法

    通常对于几何曲面的绘制采用采样描点的方式 比如球的方程为 对于这个方程可以将其改为参数方程 然后对与进行离散化 从而得到一系列的离散点 下面是c 的代码生成一个球 class Sphere public Mesh public Sphere

热门标签