使用十二面体画球体

当做一个地球的时候，可以使用六面体，八面体和20面体来画一个球体，它们的算法都是相同的，下面是20面体画球体的代码

欧拉定理(欧拉公式) V + F- E = 2 (简单多面体的顶点数 V,棱数 E 和面数 F).

1、这种方法的原理是：

　　对每一个三角面进行一次

 public GameObject GetSphere(int subdivision = 5, float radius = 1.0f)
    {
        GameObject go = new GameObject ("Pentagon");
        MeshFilter filter = go.AddComponent<MeshFilter> ();
        
        Mesh mesh = new Mesh();
        filter.mesh = mesh;
        List<Vector3> vertices = new List<Vector3>()
        {
            new Vector3(0.000000f, -1.000000f, 0.000000f),
            new Vector3(0.723600f, -0.447215f, 0.525720f),
            new Vector3(-0.276385f, -0.447215f, 0.850640f),
            new Vector3(-0.894425f, -0.447215f, 0.000000f),
            new Vector3(-0.276385f, -0.447215f, -0.850640f),
            new Vector3(0.723600f, -0.447215f, -0.525720f),
            new Vector3(0.276385f, 0.447215f, 0.850640f),
            new Vector3(-0.723600f, 0.447215f, 0.525720f),
            new Vector3(-0.723600f, 0.447215f, -0.525720f),
            new Vector3(0.276385f, 0.447215f, -0.850640f),
            new Vector3(0.894425f, 0.447215f, 0.000000f),
            new Vector3(0.000000f, 1.000000f, 0.000000f),
        };
        int[] triangles =
        {
            0, 1, 2,
            1, 0, 5,
            0, 2, 3,
            0, 3, 4,
            0, 4, 5,
            1, 5, 10,
            2, 1, 6,
            3, 2, 7,
            4, 3, 8,
            5, 4, 9,
            1, 10, 6,
            2, 6, 7,
            3, 7, 8,
            4, 8, 9,
            5, 9, 10,
            6, 10, 11,
            7, 6, 11,
            8, 7, 11,
            9, 8, 11,
            10, 9, 11
        };

        for (int s = 0; s < subdivision; s++)
        {
            
            //这将存储连接中已经存在的所有顶点，因此不存在重复/重叠的顶点
            List<VertOnEdge> connected = new List<VertOnEdge>();

            //细分后的面数
            int[] newTriangles = new int[triangles.Length * 4];
            //遍历三角形面数
            for (int i = 0; i < triangles.Length / 3; i++)
            {
                //获取一个面的三个顶点
                int A = triangles[i * 3];
                int B = triangles[i * 3 + 1];
                int C = triangles[i * 3 + 2];

                int ab = -1;
                int bc = -1;
                int ca = -1;
                List<int> connectionToBeRemoved = new List<int>(); //需要删除的连接

                for (int j = 0; j < connected.Count; j++)
                {
                    VertOnEdge voe = connected[j];
                    if (voe.A == A)
                    {
                        if (voe.B == B)
                        {
                            ab = voe.vertIndex;
                            connectionToBeRemoved.Add(j);
                        }
                        else if (voe.B == C)
                        {
                            ca = voe.vertIndex;
                            connectionToBeRemoved.Add(j);
                        }
                    }
                    else if (voe.A == B)
                    {
                        //check if there is a connection between B and A or C
                        if (voe.B == A)
                        {
                            ab = voe.vertIndex;
                            connectionToBeRemoved.Add(j);
                        }
                        else if (voe.B == C)
                        {
                            bc = voe.vertIndex;
                            connectionToBeRemoved.Add(j);
                        }
                    }
                    else if (voe.A == C)
                    {
                        //check if there is a connection between C and A or B
                        if (voe.B == A)
                        {
                            ca = voe.vertIndex;
                            connectionToBeRemoved.Add(j);
                        }
                        else if (voe.B == B)
                        {
                            bc = voe.vertIndex;
                            connectionToBeRemoved.Add(j);
                        }
                    }
                }

                connectionToBeRemoved.Sort();
                connectionToBeRemoved.Reverse();
               
                for (int k = 0; k < connectionToBeRemoved.Count; k++)
                {
                    connected.RemoveAt(connectionToBeRemoved[k]);
                }
            

            //create new vertices and connections that don't exist
                if (ab == -1)
                {
                    vertices.Add((vertices[A] + vertices[B]) / 2);
                    ab = vertices.Count - 1;
                    connected.Add(new VertOnEdge(ab, A, B));
                }

                if (bc == -1)
                {
                    vertices.Add((vertices[B] + vertices[C]) / 2);
                    bc = vertices.Count - 1;
                    connected.Add(new VertOnEdge(bc, B, C));
                }

                if (ca == -1)
                {
                    vertices.Add((vertices[C] + vertices[A]) / 2);
                    ca = vertices.Count - 1;
                    connected.Add(new VertOnEdge(ca, C, A));
                }
                //4个三角形 12个顶点
                int triangleStartingIndex = i * 12;
                //在写入三角形的时候注意要顺时针写（opengl标准），逆时针写的时候面是反的
                newTriangles[triangleStartingIndex] = A;
                newTriangles[triangleStartingIndex + 1] = ab;
                newTriangles[triangleStartingIndex + 2] = ca;

                newTriangles[triangleStartingIndex + 3] = ab;
                newTriangles[triangleStartingIndex + 4] = B;
                newTriangles[triangleStartingIndex + 5] = bc;

                newTriangles[triangleStartingIndex + 6] = bc;
                newTriangles[triangleStartingIndex + 7] = C;
                newTriangles[triangleStartingIndex + 8] = ca;

                newTriangles[triangleStartingIndex + 9] = ab;
                newTriangles[triangleStartingIndex + 10] = bc;
                newTriangles[triangleStartingIndex + 11] = ca;
            }
            for (int i = 0; i < vertices.Count; i++)
            {
                vertices[i] = vertices[i].normalized;
            }
            triangles = null;
            triangles = newTriangles;
        }

        mesh.vertices = vertices.ToArray();
        mesh.triangles = triangles;
        mesh.RecalculateNormals();
        Material material = new Material (Shader.Find ("Diffuse"));
        material.SetColor ("_Color", Color.yellow);
 
        MeshRenderer renderer = go.AddComponent<MeshRenderer> ();
        renderer.sharedMaterial = material;

        return go;



    }

2、原理：

　　先对AB和BC这两个面进行N次切分，然后按照顺序获取所有的切分后的顶点，相对于上面的算法优化了切分时的算法，比较推荐

 public GameObject GetSphere(int subdivision = 5)
    {
        GameObject go = new GameObject("Planet");

        MeshFilter filter = go.AddComponent<MeshFilter>();
        
        Mesh mesh = new Mesh();
        filter.mesh = mesh;
        
        List<Vector3> vertices = new List<Vector3>()
        {
            new Vector3(0.000000f, -1.000000f, 0.000000f),
            new Vector3(0.723600f, -0.447215f, 0.525720f),
            new Vector3(-0.276385f, -0.447215f, 0.850640f),
            new Vector3(-0.894425f, -0.447215f, 0.000000f),
            new Vector3(-0.276385f, -0.447215f, -0.850640f),
            new Vector3(0.723600f, -0.447215f, -0.525720f),
            new Vector3(0.276385f, 0.447215f, 0.850640f),
            new Vector3(-0.723600f, 0.447215f, 0.525720f),
            new Vector3(-0.723600f, 0.447215f, -0.525720f),
            new Vector3(0.276385f, 0.447215f, -0.850640f),
            new Vector3(0.894425f, 0.447215f, 0.000000f),
            new Vector3(0.000000f, 1.000000f, 0.000000f)
        };
        int[] triangles =
        {
            0, 1, 2,
            1, 0, 5,
            0, 2, 3,
            0, 3, 4,
            0, 4, 5,
            1, 5, 10,
            2, 1, 6,
            3, 2, 7,
            4, 3, 8,
            5, 4, 9,
            1, 10, 6,
            2, 6, 7,
            3, 7, 8,
            4, 8, 9,
            5, 9, 10,
            6, 10, 11,
            7, 6, 11,
            8, 7, 11,
            9, 8, 11,
            10, 9, 11
        };

          if (subdivision <= 0)
            {
                subdivision = 0;
            }
    
            int cut = (subdivision + 1);
    
            List<Vector3> newvertices  =  new List<Vector3>();
            List<int> trangles = new List<int>();
            
            Dictionary<Vector3,int> verticeidxs = new Dictionary<Vector3, int>();
            
            //对每个三角面进行遍历
            for (int i = 0; i < (triangles.Length/3); i++)
            {
                //获取三角面的顶点
                int index = i * 3;
                var a = vertices[triangles[index]];
                var b = vertices[triangles[index+1]];
                var c = vertices[triangles[index+2]];
                var ab = (b - a) / cut;//
                var ac = (c - a) / cut;
                
                List<int> newpsindex = new List<int>();
                //进行切分的次数
                for (int j = 0; j <= cut; j++)
                {
                    //切分后AB上的向量长度
                    var abcut = ab * j;
                    //
                    for (int k = 0; k <= (cut- j); k++)
                    {
                        //获取三角形上所有的顶点位置
                        var pos = a + (abcut + ac * k);
                        //新的顶点的个数
                        int newindex = newvertices.Count;
                        //如果顶点在三条边上
                        if (j == 0 || k == 0 || ((j + k) == cut))
                        {
                            //如果顶点字典中不存在该顶点
                            if (!verticeidxs.ContainsKey(pos))
                            {
                                //将该顶点添加到字典和新顶点列表中
                                newindex = newvertices.Count;
                                newvertices.Add(pos);  
                                verticeidxs.Add(pos,newindex);
                            }
                            else
                            {
                                //将字典中的该顶点的下标赋
                                newindex = verticeidxs[pos];
                            }
                        }
                        else
                        {
                            newindex = newvertices.Count;
                            newvertices.Add(pos);
                            verticeidxs.Add(pos,newindex);
                            
                        }
                        //添加新的顶点的下标
                        newpsindex.Add(newindex);
                    }
                }
    
                var s0 = 0;//下标最小数
                //组成三角面，对切分次数进行遍历，j代表第几行
                for (int j = 0; j < cut; j++)
                {
                    //下一行的第一个顶点的下标
                    var s1 = s0 + cut + 1 - j ;
                    //k代表第J行的第K个点
                    for (int k = 0; k < (cut - j); k++)
                    {
                        //添加每一行的正三角
                        trangles.Add(newpsindex[s0 + k]);
                        trangles.Add(newpsindex[s1 + k]);
                        trangles.Add(newpsindex[s0 + k + 1]);
                        //添加每一行的倒立的三角
                        if (j < cut - 1 && k<(cut - j - 1))
                        {
                            trangles.Add(newpsindex[s0 + k + 1]);
                            trangles.Add(newpsindex[s1 + k]);
                            trangles.Add(newpsindex[s1 + k + 1]);
                        }
                    }
                    s0 = s1;
                }
            }
            //将顶点归一化
            for (int i = 0; i < newvertices.Count; i++)
            {
                newvertices[i] = newvertices[i].normalized * 1;
            }
            
            vertices = newvertices;
            MeshRenderer renderer = go.AddComponent<MeshRenderer>();
            renderer.material.shader = Shader.Find("Diffuse");
            renderer.material.SetColor("_Color",Color.yellow);
            mesh.vertices = newvertices.ToArray();
            mesh.triangles = trangles.ToArray();
        

        return go;
    }